Augmenting the intertemporal CAPM with inflation: Further evidence from alternative models

Abstract

Studies consistently find that inflation is an important augmented factor for intertemporal capital asset pricing models (ICAPMs) when pricing the Fama–French 25 size and book-to-market portfolios. We extend this line of research by investigating two alternative ICAPM models (from Michel; Hahn and Lee) and the three-factor model from Hou et al. We find significant evidence that both ICAPMs and Hou et al.’s three-factor model perform better when augmented with inflation than the original models. The augmented models achieve a good model fit with the fewest factors, thus avoiding or alleviating the over-fitting problem.

Keywords

Asset pricing ICAPM inflation over-fitting problem

1. Introduction

Fama and French (1992, 1993, 1996) develop a three-factor intertemporal capital asset pricing model (ICAPM) that regards the smb and hml factors as state variables in Merton’s (1973) ICAPM framework.¹ A number of articles explore the role of various economic variables related to the smb and hml factors in the ICAPM (see, for example, Hahn and Lee, 2006; Petkova, 2006; Vassalou, 2003; Vassalou and Xing, 2004). For example, Hahn and Lee (2006) find evidence not only that the default and term spreads are good candidates for state variable proxies in the context of the Merton (1973) ICAPM but also that the innovation in the default spread and term spread may also proxy for smb and hml. In addition, Michel (2009) suggests that the real estate factor captures most of the information related to smb and hml. In other words, both Hahn and Lee’s (2006) ICAPM and Michel’s (2009) ICAPM perform as well as the Fama–French three-factor model (FF3) in pricing the cross section of the Fama–French 25 size and book-to-market (hereafter, SBM25) portfolios.

Recently, Shi et al. (2015) found that inflation² provides additional important information that both the FF3 and Petkova’s (2006) ICAPM fail to explain. This macro factor is not arbitrarily chosen in the arbitrage pricing theory (APT) model of Ross (1976), as myriad APT models commonly include the inflation factor, beginning with Chen et al. (1986). In addition, there is well-documented evidence that the inflation factor is critical for pricing asset returns (Adams et al., 2004; Bottazzi and Corradi, 1991). Maio (2013) also illustrates that inflation as a conditional variable can help improve the performance of Campbell and Vuolteenaho’s (2004) model in cross-sectional tests.

There are at least three theoretical arguments for the importance of the role of the inflation factor. First, the Fisher effect argues that the ex ante return on an asset can always be broken down into two components. The first is the real rate of return, and the other is the expected inflation rate, which suggests that inflation is always a determinant of asset returns, including stock returns. Second, Fama and Schwert (1977) also emphasize that inflation can predict aggregate stock returns. Third, Maio (2013) argues that the consumer price index (CPI) is a proxy for the “bad times” of financial wealth and nonfinancial wealth. Many studies consistently find a negative relationship between stock returns and inflation. For example, Fama (1981) documents a negative relationship between inflation and real activity, in which the real activity can be the primary driver of the stock price. As a result, the negative relationship between stock returns and inflation is also a proxy for the positive relationship between real activity and stock returns.

Apart from the inflation rate, the asset pricing literature has also considered both unexpected inflation (see, for example, Aretz et al., 2010; Chen et al., 1986) and innovation in expected inflation (see Chen et al., 1986). However, all three alternative factors are highly correlated with one another, which implies that they share a significant amount of similar information. Our unreported findings show that, empirically, the inflation rate is the best choice of the three to represent an augmented factor to capture the most valuable information. In other words, our proposed ICAPM models augmented with inflation all perform better overall than models augmented with unexpected inflation or innovation in expected inflation in terms of the value of R² and Hansen–Jagannathan (HJ)-distance and our p-value approach.

Our study extends Shi et al. (2015) and specifically investigates other ICAPM specifications. Thus, the following two alternative ICAPM specifications and another multi-factor model are chosen as benchmark models in this study: Michel’s (2009) ICAPM, Hahn and Lee’s (2006) ICAPM, and Hou et al.’s (2011) model. Notably, these ICAPM specifications are selected with the following restrictive requirement: the state variable in these ICAPM specifications must contain most of the information of the smb and hml factors in pricing the SBM25 portfolios.³ Michel (2009) shows that innovations in real estate portfolios meet this requirement, whereas Hahn and Lee (2006) illustrate that the smb and hml factors may proxy for innovations in the default and term spreads, respectively. Hou et al. (2011) find that their three-factor model that enhances the CAPM with momentum and cash flow-to-price (CFP) factors outperforms the FF3 and an alternative version of the Carhart (1997) four-factor model in explaining the time series of global stock returns.

Our methodology is based on Fama and MacBeth’s (1973) two-pass ordinary least squares (OLS) and generalized least squares (GLS) cross-sectional regression and Hansen’s (1982) generalized method of moments (GMM). Following Kan et al. (2013), we also report the p-value of the R² statistic, the p-value of a generalized version of Shanken’s (1985) F-test statistics, and the standard error of R² under the assumption that 0 < R² < 1 in both OLS and GLS estimations. Both statistics mentioned above ease the independent and identically distributed (i.i.d.) normality assumption. The p-value of the R² statistic aims to test R² = 1 (i.e. a correctly specified model) and R² = 0 (i.e. a misspecified model that explains none of the cross-sectional variation in the expected returns). In addition, we conduct a robustness check that adds five Fama–French industry portfolios to the SBM25 portfolios to ease the tight structure of the SBM25 portfolios, as recommended by Lewellen et al. (2010) and Kan et al. (2013).

Using US monthly data over the period spanning from July 1963 to September 2014, we find significant evidence that the alternative ICAPMs augmented with inflation perform better than their original benchmark. Apart from having higher R² statistics and lower pricing errors, the ICAPMs augmented with inflation also pass the p-value approach, whereas the original ICAPMs fail in this regard, highlighting the fact that inflation (the augmented factor) is an important factor that is commonly regarded in both APT and multi-factor models and that helps the original ICAPMs and HKK model improve their performance in pricing the SBM25 and the SBM25 plus five industry portfolios.

Our contributions to the literature are as follows. First, consistent with Shi et al. (2015), we find that the explanatory power of the benchmark ICAPMs is inadequate; for example, the data do not support the null hypothesis that the model is correctly specified in our GLS or GMM estimations for Hahn and Lee’s (2006) model, while Michel’s (2009) model performs poorly in various model diagnostic tests. The HKK model performs inadequately as well. Second, our empirical evidence clearly suggests that the alternative inflation-augmented models perform significantly better than the original ICAPMs in explaining the cross section of the SBM25 portfolios. We can alternatively regard our augmented multi-factor models as Ross’ (1976) multi-factor APT models.⁴ In particular, Armitage (2005) also posits that the APT model is practically indistinguishable from an ad hoc multi-factor model.

Asset pricing in the Australian market also has a long period of fruitful investigation. For examples, Faff and Chan (1998) test an ICAPM model with a gold bullion price factor as a hedging variable; Docherty et al. (2010) propose a tangibility factor; Li et al. (2011) document the usefulness of the consumption factor; and Chai et al. (2013) use a liquidity factor. All these related studies investigate certain new factors/models to explain the variation in Australian stock market returns in some regard. However, to the best of our knowledge, none of these studies clearly answers the question of whether the inflation factor helps the original models price the cross section of equity market returns. The success of the US market study in this article may shed some light on this point, that is, that the inflation factor may enhance the explanatory power of the original models in the Australian market and in other international markets.⁵

The remainder of the article proceeds as follows. Section 2 outlines our inflation-augmented ICAPMs. Section 3 discusses the data and the underlying econometrics. Section 4 presents the empirical results, and section 5 concludes.

2. The proposed benchmark models

The first set of our proposed alternative models is Michel’s (2009) ICAPM augmented with inflation, and the second set of alternative models is Hahn and Lee’s (2006) ICAPM model augmented with inflation. The state variables in Hahn and Lee’s (2006) ICAPM are the innovations in the term spread and the default spread, while the state variable in Michel’s (2009) ICAPM is the innovation in real estate portfolios. Campbell (1996) argues that implementations of the ICAPM should not rely solely on choosing important macroeconomic variables. Instead, the factors in the model should be related to the innovations in state variables that forecast future investment opportunities. In light of Campbell (1996), Hahn and Lee (2006) and Michel (2009) both argue that the term spread, default spread, and real estate portfolio are capable of forecasting future market returns and are therefore good candidates for state variables in the ICAPM framework.

We choose Michel’s (2009) ICAPM and Hahn and Lee’s (2006) ICAPM formulations mainly because the state variables in these two ICAPMs capture most of the implications of the smb and hml factors in pricing the SBM25 portfolios. The following equations represent the beta-pricing form of our proposed ICAPMs augmented with inflation.

The first proposed model is Michel’s (2009) ICAPM augmented with inflation

r_{i, t} = β_{r m r f} r m r f_{t} + β_{s m b} r e a l e s t a t e_{t} + β_{i n f} i n f_{t} + ε_{i, t}

(1)

where rmrf is the market risk premium, that is, the difference between the value-weighted market return and the 1-month Treasury bill rate; inf is the inflation, that is, the first difference of the logarithmic value of CPI; and r is the excess return on the SBM25 portfolios. $β$ in any notation refers to the parameter and $ε_{i, t}$ refers to the pricing error.

The second proposed ICAPM is Hahn and Lee’s (2006) ICAPM augmented with inflation

r_{i, t} = β_{r m r f} r m r f_{t} + β_{γ^{t e r m}} γ_{t}^{t e r m} + β_{γ^{d e f}} γ_{t}^{d e f} + β_{i n f} i n f_{t} + ε_{i, t}

(2)

where $γ_{t}^{x}$ is the innovation in variable x (and x is one of the state variables); term is the term spread, that is, the difference between the yields of a 10-year and a 1-year government bond; and def is the default spread, that is, the difference between the yields of Moody’s Baa and Aaa corporate bond yields.

In addition to these ICAPMs, we propose Hou et al.’s (2011) three-factor model (HKK model) augmented with inflation

r_{i, t} = β_{r m r f} r m r f_{t} + β_{u m d} u m d + β_{c f p} c f p + β_{i n f} i n f_{t} + ε_{i, t}

(3)

where umd is the momentum premium, that is, the difference in returns between high prior return and low prior return firms; and cfp is the CFP, that is, the highest CFP-quintile returns minus the lowest CFP-quintile returns.

We augment only one essential factor with respect to Michel’s (2009) ICAPM, Hahn and Lee’s (2006) ICAPM and the HKK model; our empirical findings suggest that our augmented models have already adequately explained the cross-sectional variation in the Fama–French portfolio returns, as all the diagnostic test statistics in our study do not reject the null hypothesis that the model is correctly specified and the R² statistics are very high.

Another advantage of our proposed models is that they can alleviate the over-fitting problem in pricing the SBM25 portfolios. The over-fitting problem may lead to another problem in which the stochastic discount factor (SDF) can often be negative,⁶ violating asset pricing theory because the theory assumes the SDF to be strictly positive. To successfully explain the variation while minimizing the over-fitting problem, a good model with fewer factors is preferred. To alleviate this issue, we propose our augmented alternative models with as few factors as possible to avoid the over-fitting problem while continuing to have adequate power to explain the variation in the cross section of SBM25 portfolios.

3. Methodology and data

Our methodology is based on Fama and MacBeth’s (1973) two-pass OLS and GLS cross-sectional regression and Hansen’s (1982) GMM. In the first stage of the Fama–MacBeth regression, we obtain the estimated loadings of risk factors.⁷ In the second step, we perform cross-sectional regressions and use Shanken’s (1992) procedure to make asymptotically valid error-in-variable adjustments of the resulting standard errors.

Lewellen et al. (2010) argue that asset pricing tests are often misleading when they rely purely on the metrics of high cross-sectional R² statistics and small pricing errors.⁸ Thus, these authors recommend the p-value approach, in which the null hypothesis is that the model is correctly specified. A correctly specified model always performs better than a misspecified model in the p-value approach for every statistic. Based on this approach, Kan et al. (2013) develop the p-value test of the R² statistic.⁹ In contrast to Lewellen et al. (2010), our study compares the original ICAPMs and ICAPMs with inflation based on the p-value approach for each test statistic,¹⁰ in addition to the R² statistics and pricing errors. Second, we also report the results of Fama and MacBeth’s (1973) GLS and GMM (using the HJ-distance as a weighting matrix) to effectively alleviate this problem, as advocated by Lewellen et al. (2010). Note that the HJ-approach/two-step optimal GMM is an alternative to GLS/weighted least squares (WLS) cross-sectional regressions. In addition, we conduct a robustness check that adds five Fama–French industry portfolios to the SBM25 portfolios to relax the tight structure of the SBM25 portfolios, as recommended by Lewellen et al. (2010) and Kan et al. (2013).

We report the R² statistics as per Kan et al. (2013), Lewellen et al. (2010), and Kandel and Stambaugh (1995). Following Kan et al. (2013), we also report the p-value of the R² statistic, the p-value of the generalized version of Shanken’s (1985) F-test statistics, and the standard error of R² under the assumption that 0 < R² < 1 in both the OLS and GLS estimations. Both statistics mentioned above relax the i.i.d. normality assumption. Following Kan et al. (2013) and Davies et al. (2015), we report the results by setting the lags equal to 0, which is equivalent to a heteroskedasticity-robust adjustment. The p-value of the R² statistic aims to test R² = 1 (i.e. a correctly specified model) and R² = 0 (i.e. a misspecified model that does not explain any of the cross-sectional variation in the expected returns). In addition, we follow Campbell and Vuolteenaho (2004) to report an alternative R² statistic when the intercept is constrained to 0. This alternative R² statistic provides additional valuable information¹¹ regarding the explanatory power of asset pricing models for the cross-sectional variation in portfolios returns.

As a robustness check, we further obtain supplemental estimations using Hansen’s GMM methodology with both Hansen and Jagannathan’s (1997) weighting matrix¹² and an asymptotically optimal weighting matrix.¹³ In the SDF framework, Cochrane (2005: 253) argues that the pricing error $g_{t} = E (m R) - 1$ is simply a set of moment conditions, where R denotes the gross return of each portfolio and m is the pricing kernel. The GMM approach minimizes the weighted average value of moment conditions with any selected weighting matrix: $J (q) = [{\bar{g} {(b)}_{T}}^{T} W {\bar{g}}_{T} (b)]$ , where W is a weighting matrix. We also compute Hansen’s J-statistic for the model over-identifying restrictions using an asymptotically optimal weighting matrix in which the first stage is either an HJ-based or an identity-weighting matrix.

Our p-value approach is based on the following test statistics of model diagnostics: the R² statistic (R² = 1) and the generalized Shanken’s (1985) F-test statistics in the Fama–MacBeth regression, and the HJ-distance and Hansen’s J-statistic in the GMM. The null hypotheses of all these statistics are that the model is correctly specified. Thus, a high p-value indicates that the model is “acceptable.” Notably, Lewellen et al. (2010) propose the p-value approach in judging whether a model is correctly specified or not. We also obtain the other p-value of the R² statistic (R² = 0) in the Fama–MacBeth regression and the Wald test of the GMM. When the original model is compared with one that is augmented with a factor, the p-value approach can describe how much this factor improves the performance of the original model.¹⁴ Note that aside from R², the HJ-distance, J-test, and Shanken’s (1985) F-statistic can all measure the performance of the models.

Our sample consists of US monthly data over the period from July 1963 to September 2014. Sourced from Kenneth French’s website,¹⁵ we obtain the rate of return information of Fama and French’s (1993) SBM25s portfolio, as well as the premiums on market risk (rmrf) and the 1-month Treasury bill rate. We use a 1-month Treasury bill as a proxy for the risk-free asset. Data on the term spread, default spread, CPI, and industrial production index come from the FRED database of the Federal Reserve Bank of St Louis. The term spread is computed as the difference between the yields of a 10-year and a 1-year government bond, whereas the default spread is computed as the difference between the yields of Moody’s Baa and Aaa corporate bonds. Inflation is calculated as the first difference of log CPI in each month.

We follow Michel’s (2009) approach in constructing the real estate portfolio.¹⁶ In particular, we use the composite Real Estate Investment Trusts (REIT) index as the proxy for the real estate portfolio and compute the return on the real estate portfolio as the difference between the returns on the composite REIT index and the 1-month Treasury bill rate in our analysis. We then estimate Michel’s (2009) ICAPM and our augmented Michel’s (2009) model using the innovations of real estate portfolios. The data of the real estate index are sourced from the National Association of Real Estate Investment Trusts (NAREIT) website (www.nareit.com). Note that the data regarding the return of the real estate index are only available beginning in January 1972.

To obtain the innovations of state variables, we follow Hahn and Lee (2006) and Maio and Santa-Clara (2012) to process the data. In particular, the innovation of each state variable represents the first difference of an original variable, $γ_{t}^{x} = x_{t} - x_{t - 1}$ , where $x$ is the state variable “term, def” in Hahn and Lee’s (2006) ICAPM or the state variable “realestate” in Michel’s (2009) ICAPM.

Following Hou et al. (2011), we construct a CFP factor-mimicking portfolio. Our CFP mimicking portfolio returns are calculated as the highest CFP-quintile returns minus the lowest CFP-quintile returns, where the monthly CFP portfolio returns are from Kenneth French’s website. The momentum factor (umd) is also from Kenneth French’s website.

Table 1 summarizes the data statistics. Notably, the inflation rate has the highest autoregressive value (0.62), and other variables generally have first-order autoregressive coefficients of less than 0.40.

Table 1.

Summary statistics.

This table reports the mean, standard deviation, minimum value, maximum value and first-order autocorrelation of key variables for the sample. rmrf = market risk premium, the difference between the value-weighted market return and the 1-month Treasury bill rate; inf = inflation based on the consumer price index; term = term spread, the difference between the yields of a 10-year and a 1-year government bond; def = default spread, the difference between the yields of Moody’s Baa and Aaa corporate bonds; realestate = real estate portfolio, the difference between the return of the composite REITs index and the 1-month Treasury bill rate; umd = momentum premium, the difference of returns between high prior return and low prior return firms; cfp = cash flow-to-price, the highest-CFP-quintile returns minus the lowest-CFP-quintile returns; $γ^{x}$ = innovation in state variable x. rmrf, inf, term, def, umd, and cfp are from July 1963 to September 2014, while realestate is from January 1972 to September 2014.

	Observations	Mean	Standard deviation	Minimum	Maximum	ρ(1)
rmrf	615	0.50	4.47	−23.24	16.10	0.07
$γ^{d e f}$	615	0.00	0.12	−0.63	0.94	0.29
$γ^{t e r m}$	615	0.00	0.28	−1.56	2.62	0.30
$γ^{r e a l e s t a t e}$	512	−0.01	6.99	−35.27	40.03	−0.40
inf	615	0.33	0.32	−1.79	1.79	0.62
umd	615	0.68	4.23	−34.70	18.39	0.06
cfp	615	0.38	3.38	−18.47	14.97	0.03

Table 2 shows that the correlations between inflation and other factors are low, which implies that inflation may contain information beyond those factors in our original model specifications. The ICAPM statistics summarized in Tables 1 and 2 are consistent with Shi et al. (2015).

Table 2.

Correlation matrix.

This table reports the correlation matrix of variables in our proposed models. rmrf = market excess return; term = term spread, the difference between the yields of a 10-year and a 1-year government bond; def = default spread, the difference between the yields of Moody’s Baa and Aaa corporate bonds; inf = inflation based on the consumer price index; umd = momentum premium, the difference of returns between high prior return and low prior return firms; cfp = cash flow-to-price, the highest-CFP-quintile returns minus the lowest-CFP-quintile returns; $γ^{x}$ = innovation in state variable x, where x is term, def, realestate. Sample period is from July 1963 to September 2014 for Panels A and C, and from January 1972 to September 2014 for Panel B, respectively.

Panel A: Hahn and Lee’s ICAPM with inflation
	rmrf	$γ^{d e f}$	$γ^{t e r m}$
$γ^{d e f}$	−0.05
$γ^{t e r m}$	0.12	0.14
inf	−0.12	−0.06	0.03
Panel B: Michel’s ICAPM with inflation
	rmrf	$γ^{r e a l e s t a t e}$
g^realestate	0.38
inf	−0.11	−0.06
Panel C: HKK model with inflation
	rmrf	umd	cfp
umd	−0.12
cfp	−0.23	−0.07
inf	−0.12	0.08	0.07

ICAPM: intertemporal capital asset pricing models.

4. Empirical results

4.1. The preliminary evidence

Table 3 clearly illustrates that the risk premium of inflation is significant in two-pass OLS and GLS cross-sectional regressions, suggesting that inflation is a significant risk factor that helps price the SBM25 portfolios. Moreover, the R² statistic of Kan et al. (2013) in OLS and GLS is 0.88 and 0.37, respectively.

Table 3.

The significant role of inflation in 25 size and book-to-market portfolios.

This table reports the estimates of the single factor inflation from Fama–MacBeth’s two-pass OLS and GLS regressions for the SBM25 portfolios. Following Kan et al. (2013), we report the OLS cross-sectional R² (in %). The sample period is from July 1963 to September 2014.

OLS estimation
	inf		OLS
Coefficient	−0.38	R ²	0.88
t-stat. (fm)	−3.47
t-stat. (s)	−2.21
GLS estimation
	inf		GLS
Coefficient	−0.34	R ²	0.37
t-stat. (fm)	−6.84
t-stat. (s)	−4.73

OLS: ordinary least squares; inf: inflation based on the consumer price index; GLS: generalized least squares.

4.2. Fama–MacBeth’s approach

We display the results of Fama–MacBeth’s two-pass OLS and GLS regressions in Tables 4 and 5, respectively. We show the performance of our proposed ICAPMs augmented with inflation, in addition to the performance of the original specifications. In addition to the R² statistic and pricing error, we use the p-value approach to compare the performance of the models. We restrict the zero-beta rate to be equal to the risk-free rate, and the intercepts in both OLS and GLS are thus constrained to be 0. As is typical for a regression analysis without a constant, we follow Kan et al. (2013) to report a non-negative R² value. This non-negative R² statistic is always quite high in Fama–MacBeth’s two-pass OLS regression; for example, the CAPM model achieves a 0.86 R² value in Kan et al. (2013). In Fama–MacBeth’s two-pass OLS regression, augmenting the models with the inflation factor always slightly increases the value of Kan et al.’s (2013) R² and decreases the value of the standard error of the R². For one example in Table 4, the R² and its standard error of Michel’s (2009) ICAPM is 0.96 and 0.03, respectively, while the R² and its standard error of Michel’s (2009) ICAPM augmented with inflation is 0.97 and 0.02, respectively. Aside from the p-value approach, which can provide more robust evidence of the extent of the improvement provided by our inflation-augmented models, the alternative R² of Campbell and Vuolteenaho (2004) provides more valuable information. All the inflation-augmented models have an increase of approximately 17% or 20% in the R² of Campbell and Vuolteenaho (2004), also confirming that the augmented models perform better.

Table 4.

25 Size and book-to-market portfolios: OLS estimation results.

This table reports the estimates of risk premium from Fama–MacBeth’s two-pass OLS regressions for the SBM25 portfolios. rmrf = market excess return; term = term spread, the difference between the yields of a 10-year and a 1-year government bond; def = default spread, the difference between the yields of Moody’s Baa and Aaa corporate bonds; realstate = real estate portfolio, the difference between the returns on the composite REITs index and 1-month Treasury bill rate; inf = inflation based on the consumer price index; $γ^{x}$ = innovation in variable x, where x is term, def, realestate; umd = momentum premium, the difference of returns between high prior return and low prior return firms; cfp = cash flow-to-price, the highest-CFP-quintile returns minus the lowest-CFP-quintile returns; t-stat. (fm) = the Fama–MacBeth t-statistics; t-stat. (s) = the Shanken (1992) error-in-variable adjusted t-statistics; p(R² = 1) and p(R² = 0) are p-value of R² in testing R² = 1 and R² = 0, respectively. p(F-test) is the p-value of F-test. The p-value of F-test is the p-value of a generalized version of Shanken’s (1985) cross-sectional regression test (CSRT), following Kan et al. (2013). Following Kan et al. (2013), we report OLS cross-sectional R² (in %) and corresponding p-value, and rse is the standard error of R² under the assumption that 0 < R² < 1. &R² is an alternative OLS R², following Campbell and Vuolteenaho (2004). The sample period is from January 1972 to September 2014 for Michel’s (2009) ICAPM and its inflation-augmented model, while the sample period is from July 1963 to September 2014 for other models.

OLS estimation of model
Michel’s ICAPM
	rmrf	$γ^{r e a l e s t a t e}$	inf			OLS
Coefficient	0.39	4.34			R ²	0.96	&R²	0.51
t-stat. (fm)	1.84	4.81			p(R² = 1)	0.07	p(F-test)	0.00
t-stat. (s)	1.80	4.12			p(R² = 0)	0.00	rse	0.03
Michel’s ICAPM + inf
	rmrf	$γ^{r e a l e s t a t e}$				OLS
Coefficient	0.43	4.57	−0.27		R ²	0.97	&R²	0.68
t-stat. (fm)	2.07	4.99	−2.29		p(R² = 1)	0.33	p(F-test)	0.25
t-stat. (s)	1.98	3.56	−1.59		p(R² = 0)	0.00	rse	0.02
Hahn and Lee’s ICAPM
	rmrf	$γ^{d e f}$	$γ^{t e r m}$			OLS
Coefficient	0.54	0.04	0.47		R ²	0.97	&R²	0.63
t-stat. (fm)	2.93	1.13	4.96		p(R² = 1)	0.45	p(F-test)	0.36
t-stat. (s)	2.72	0.58	2.53		p(R² = 0)	0.00	rse	0.02
Hahn and Lee’s ICAPM + inf
	rmrf	$γ^{d e f}$	$γ^{t e r m}$	inf		OLS
Coefficient	0.53	0.00	0.44	−0.37	R ²	0.98	&R²	0.80
t-stat. (fm)	2.86	0.05	4.58	−5.40	p(R² = 1)	0.96	p(F-test)	0.93
t-stat. (s)	2.58	0.02	2.05	−2.43	p(R² = 0)	0.00	rse	0.015
HKK model
	rmrf	umd	cfp			OLS
Coefficient	0.51	1.89	1.00		R ²	0.96	&R²	0.52
t-stat. (fm)	2.68	1.52	4.68		p(R² = 1)	0.03	p(F-test)	0.00
t-stat. (s)	2.62	1.29	4.23		p(R² = 0)	0.00	rse	0.024
HKK model + inf
	rmrf	umd	cfp	inf		OLS
coefficient	0.54	2.20	0.95	−0.33	R ²	0.98	&R²	0.72
t-stat. (fm)	2.83	1.86	4.32	−3.08	p(R² = 1)	0.43	p(F-test)	0.58
t-stat. (s)	2.59	1.14	3.03	−1.89	p(R² = 0)	0.00	rse	0.021

REIT: Real Estate Investment Trusts; CFP: cash flow-to-price; OLS: ordinary least squares; ICAPM: intertemporal capital asset pricing models.

Table 5.

25 Size and book-to-market portfolios: GLS estimation results.

This table reports the estimates of risk premium from Fama–MacBeth’s two-pass GLS regressions for the SBM25 portfolios. rmrf = market excess return. term = term spread, the difference between the yields of a 10-year and a 1-year government bond; def = default spread, the difference between the yields of Moody’s Baa and Aaa corporate bonds; realstate = real estate portfolio, the difference between the returns on the composite REITs index and 1-month Treasury bill rate; inf = inflation based on the consumer price index; $γ^{x}$ = innovation in variable x, where x is term, def, realestate; umd = momentum premium, the difference of returns between high prior return and low prior return firms; cfp = cash flow-to-price, the highest-CFP-quintile returns minus the lowest-CFP-quintile returns; t-stat. (fm) = the Fama–MacBeth t-statistics; t-stat. (s) = the Shanken (1992) error-in-variable adjusted t-statistics; p(R² = 1) and p(R² = 0) are p-value of R² in testing R² = 1 and R² = 0, respectively. p(F-test) is the p-value of F-test. The p-value of F-test is the p-value of a generalized version of Shanken’s (1985) CSRT, following Kan et al. (2013). Following Kan et al. (2013), we report GLS cross-sectional R² (in %) and corresponding p-value, and rse is the standard error of R² under the assumption that 0 < R² < 1. The sample period is from January 1972 to September 2014 for Michel’s (2009) ICAPM and its inflation-augmented model, while the sample period is from July 1963 to September 2014 for other models.

GLS estimation of models
Michel’s ICAPM
	rmrf	$γ^{r e a l e s t a t e}$				GLS
Coefficient	0.53	2.75			R ²	0.17
t-stat. (fm)	2.63	4.07			p(R² = 1)	0.00	p(F-test)	0.00
t-stat. (s)	2.63	3.84			p(R² = 0)	0.00	rse	0.08
Michel’s ICAPM + inf
	rmrf	$γ^{r e a l e s t a t e}$	inf			GLS
Coefficient	0.51	2.14	−0.38		R ²	0.54
t-stat. (fm)	2.54	3.14	−6.71		p(R² = 1)	0.57	p(F-test)	0.28
t-stat. (s)	2.52	2.16	−4.42		p(R² = 0)	0.00	rse	0.18
Hahn and Lee’s ICAPM
	rmrf	$γ^{d e f}$	$γ^{t e r m}$			GLS
Coefficient	0.56	0.02	0.26		R ²	0.26
t-stat. (fm)	3.10	1.29	4.99		p(R² = 1)	0.00	p(F-test)	0.00
t-stat. (s)	3.09	0.95	3.65		p(R² = 0)	0.01	rse	0.17
Hahn and Lee’s ICAPM + inf
	rmrf	$γ^{d e f}$	$γ^{t e r m}$	inf		GLS
Coefficient	0.54	0.02	0.26	−0.36	R ²	0.59
t-stat. (fm)	2.98	1.15	4.85	−6.27	p(R² = 1)	0.83	p(F-test)	0.63
t-stat. (s)	2.95	0.66	2.75	−3.55	p(R² = 0)	0.00	rse	0.21
HKK model
	rmrf	umd	cfp			GLS
Coefficient	0.54	2.13	0.54		R ²	0.26
t-stat. (fm)	3.03	3.90	3.17		p(R² = 1)	0.00	p(F-test)	0.00
t-stat. (s)	3.02	3.39	2.99		p(R² = 0)	0.00	rse	0.14
HKK model + inf
	rmrf	umd	cfp	inf		GLS
Coefficient	0.51	0.93	0.50	−0.36	R ²	0.54
t-stat. (fm)	2.87	1.61	2.92	−5.90	p(R² = 1)	0.46	p(F-test)	0.46
t-stat. (s)	2.85	1.05	2.36	−3.82	p(R² = 0)	0.00	rse	0.20

GLS: generalized least squares; SBM25: 25 size and book-to-market; REITs: Real Estate Investment Trusts; CFP: cash flow-to-price; ICAPM: intertemporal capital asset pricing models.

Consider first Fama–MacBeth’s OLS results for the SBM25 portfolios in Table 4. In the estimates of Michel’s (2009) ICAPM augmented with inflation, the inflation factor is not significantly priced based on Shanken’s (1992) error-in-variable adjusted t-statistic but is significantly priced based on Fama–MacBeth’s t-statistic. Nonetheless, the p-values of the R² statistic (R² = 1) and Shanken’s F-statistic are significant, while they are insignificant in Michel’s (2009) model, suggesting that our augmented model significantly outperforms Michel’s (2009) model. On the other hand, comparing the original Hahn and Lee’s (2006) model with our Hahn and Lee’s (2006) inflation-augmented model in the OLS setup, the two perform almost equivalently and successfully in passing the p-value test of the R² statistic (R² = 1) and Shanken’s F-statistic, although the augmented inflation factor is significantly priced based on Shanken’s (1992) t-statistic. The GLS estimation results clearly illustrate the superiority of our augmented ICAPMs. In addition, the p-value of the R² statistic (R² = 1) and Shanken’s F-statistic reject the HKK model at the level of 0.05 significance. In the HKK augmented with inflation model, the augmented inflation is significantly priced based on Shanken’s (1992) adjusted t-statistic of −1.89, and the model also successfully passes the p-value of the R² statistic (R² = 1) and Shanken’s F-statistic at the level of 0.10 significance. The negative sign of the risk premium of inflation is consistent with the findings from Chen et al. (1986). One explanation for the negative inflation premium is that most equities have negative sensitivities to inflation risk.

We then perform Fama–MacBeth’s GLS estimations on the SBM25 portfolios and present the results in Table 5. Following the p-value approach of the R² statistic (R² = 1) and Shanken’s F-statistic, we find significantly better performance of Michel’s (2009) ICAPM augmented with inflation, Hahn and Lee’s (2006) ICAPM augmented with inflation, and the HKK model augmented with inflation. In contrast, the two original ICAPMs and the HKK model perform poorly, which demonstrates that our inflation-augmented ICAPMs and augmented HKK model are more robust than the original models in pricing the SBM25 portfolios.

Moreover, the risk premium of the inflation factor is significantly priced in the GLS estimation of our three augmented models. The estimated coefficient is −0.38 with Shanken’s (1992) t-statistic of −4.42 in Michel’s (2009) ICAPM augmented with inflation, the estimated coefficient is −0.36 with Shanken’s (1992) t-statistic of −3.55 in Hahn and Lee’s (2006) ICAPM augmented with inflation, and the estimated coefficient is −0.36 with Shanken’s (1992) t-statistic of −3.82 in the HKK augmented with inflation, implying that inflation is an important factor that further helps price the SBM25 portfolios. In Tables 4 and 5, every model passes the p-value approach of the R² statistic (R² = 0), suggesting that each model has some power in explaining the cross-sectional variation in portfolio returns. We also report the standard error of R² and find that the augmented models perform better in both OLS and GLS, which is consistent with the findings of Kan et al. (2013).

4.3. GMM estimations

Because the SDF framework is an alternative method that differs from Fama–MacBeth’s method, we conduct the GMM with both the HJ-weighting matrix (Hansen and Jagannathan, 1997) and an optimal weighting matrix as a supplementary test. We find consistent or similar evidence between the GMM estimations using these two different weighting matrixes. We prefer to report the results of the HJ-approach because estimation by the HJ-distance involves a common weighting matrix across models, thus enabling us to consistently compare the performance of the models under consideration.

Within the GMM approach, the null of the J-test¹⁷ is that the model is valid, indicating that the moment condition is correctly specified. The null hypothesis of the HJ-distance test is that the model is valid and the squared pricing errors are significantly different from 0. The high p-value of the J-test or the HJ-distance thus implies that the model has a good fit with the data.

Table 6 reports the testing results of our ICAPMs augmented with inflation and competing models using GMM. The J-test and the HJ-distance test reject Michel’s (2009) ICAPM, Hahn and Lee’s (2006) ICAPM, and the HKK model, although the Wald test statistics are all significant. These results imply that Michel’s (2009) model, Hahn and Lee’s (2006) model, and the HKK model are not correctly specified in these estimations. In contrast, our models perform consistently better. Our Michel’s (2009) ICAPM augmented with inflation passes the model diagnostics in terms of the Wald test statistic (11.98 with a p-value of 0.00), J-test statistic (28.70 with a p-value of 0.12), and HJ-distance (0.28 with a p-value of 0.49). Our Hahn and Lee’s (2006) ICAPM augmented with inflation also passes these tests in terms of the Wald test statistic (11.07 with a p-value of 0.03), J-test statistic (24.39 with a p-value of 0.23), and HJ-distance (0.26 with a p-value of 0.65). Furthermore, our HKK model augmented with inflation also passes these tests in terms of the Wald test statistic (11.48 with a p-value of 0.02), J-test statistic (26.55 with a p-value of 0.15), and HJ-distance (0.28 with a p-value of 0.28).

Table 6.

25 Size and book-to-market portfolios: GMM estimation results.

This table reports the GMM estimation results for the SBM25 portfolios using Hansen–Jagannathan distance (Hansen and Jagannathan, 1997). The HJ-distance uses the inverse of the second moments of the 25 SBM portfolio returns as the weighting matrix. rmrf = market risk premium, the difference between the value-weighted market return and the 1-month Treasury bill rate; term = term spread, the difference between the yields of a 10-year and a 1-year government bond; def = default spread, the difference between the yields of Moody’s Baa and Aaa corporate bonds; realstate = real estate portfolio, the difference between the returns on the composite REITs index and 1-month Treasury bill rate; inf = inflation based on the consumer price index; $γ_{t}^{x}$ = innovation in variable x, where x is term, def, realestate; umd = momentum premium, the difference of returns between high prior return and low prior return firms; cfp = cash flow-to-price, the highest-CFP-quintile returns minus the lowest-CFP-quintile returns. The J-test is Hansen’s (1982) test of the model over-identifying restrictions. The Wald test examines the joint significance of factor loadings in the pricing kernel. The HJ-distance test is the Hansen–Jagannathan distance measure (Hansen and Jagannathan, 1997) whose p-values are obtained from 10,000 simulations as in Jagannathan and Wang (1996). The bs are the coefficient estimates of the pricing factors and the constant in the pricing kernel, while λ is the estimate of the corresponding risk premium. t-stat. = the GMM t-statistics. The numbers in brackets are the associated p-values. The sample period is from January 1972 to September 2014 for Michel’s (2009) ICAPM and its inflation-augmented model, while the sample period is from July 1963 to September 2014 for other models.

Michel’s ICAPM
	cons	rmrf	$γ^{r e a l e s t a t e}$			Wald	J-test	HJ-dist.
b	0.95	0.05	−0.03			11.62	55.20	0.35
t-stat.	81.25	3.39	−2.05			(0.00)	(0.00)	(0.00)
λ		−0.76	1.02
t-stat.		−2.56	1.38
Michel’s ICAPM + inf
	cons	rmrf	$γ^{r e a l e s t a t e}$	inf		Wald	J-test	HJ-dist
b	0.04	0.05	−0.03	2.69		11.98	28.70	0.28
t-stat.	0.12	2.25	−1.17	2.97		(0.00)	(0.12)	(0.49)
λ		−0.31	1.17	−0.31
t-stat.		−0.65	1.00	−2.94
Hahn and Lee’s ICAPM
	cons	rmrf	$γ^{d e f}$	$γ^{t e r m}$		Wald	J-test	HJ-dist.
b	0.96	0.04	0.81	−1.90		11.17	50.44	0.33
t-stat.	44.24	2.68	0.65	−2.26		(0.01)	(0.00)	(0.00)
λ		−0.64	−0.00	0.14
t-stat.		−1.82	−0.08	2.08
Hahn and Lee’s ICAPM + inf
	cons	rmrf	$γ^{d e f}$	$γ^{t e r m}$	inf	Wald	J-test	HJ-dist.
b	−0.02	0.05	1.01	−2.63	2.98	11.07	24.39	0.26
t-stat.	−0.06	2.08	0.48	−2.26	2.79	(0.03)	(0.23)	(0.65)
λ		−0.15	0.01	0.18	−0.28
t-stat.		−0.28	0.19	2.06	−2.67
HKK model
	cons	rmrf	umd	cfp		Wald	J-test	HJ-dist.
b	1.02	0.01	−0.05	−0.03		9.96	65.29	0.34
t-stat.	23.99	0.92	−1.45	−1.68		(0.02)	(0.00)	(0.00)
λ		−0.62	1.00	0.41
t-stat.		−1.88	1.55	2.18
HKK model + inf
	cons	rmrf	umd	cfp	inf	Wald	J-test	HJ-dist.
b	0.04	0.01	−0.04	−0.05	2.97	11.48	26.55	0.28
t-stat.	0.11	0.72	−0.80	−1.73	2.93	(0.02)	(0.15)	(0.28)
λ		−0.16	0.48	0.43	−0.29
t-stat.		−0.34	0.50	1.55	−2.84

CFP: cash flow-to-price; GMM: generalized method of moments; REIT: Real Estate Investment Trusts; 25 SBM: 25 size and book-to-market; ICAPM: intertemporal capital asset pricing models.

The GMM estimation results of Michel’s (2009) ICAPM augmented with inflation show that the coefficient of inflation is 2.69 with a t-statistic of 2.97, and the risk premium of inflation is −0.31 with a t-statistic of −2.94. As for Hahn and Lee’s (2006) ICAPM augmented with inflation, the coefficient of inflation is 2.98 with a t-statistic of 2.79, and the risk premium of inflation is −0.28 with a t-statistic of −2.67, whereas for the HKK model augmented with inflation, the coefficient of inflation is 2.97 with a t-statistic of 2.93, and the risk premium of inflation is −0.29 with a t-statistic of −2.84. The above findings suggest that augmented inflation is a key macro factor that not only is significantly priced but also helps price the SBM25 portfolios.

Our unreported tests show the following evidence regarding the over-fitting problem. Petkova’s (2006) ICAPM augmented with macro-factors in Shi et al. (2015) can pass other key test statistics of diagnoses but does not perform well in the Wald test of GMM (using an identity-weighting matrix or HJ-distance), which indicates some degree of the over-fitting problem. Conversely, Michel’s (2009) ICAPM augmented with inflation and the HKK model augmented with inflation both pass the Wald test statistic successfully. Moreover, Hahn and Lee’s (2006) ICAPM augmented with inflation can alleviate the over-fitting problem, in which the Wald statistics of GMM using the identity-weighting matrix or HJ-distance are better in terms of statistical significance.

4.4. Robustness check

Finally, following Kan et al. (2013), we conduct a robustness test by adding five Fama–French industry portfolios to the Fama–French 25 portfolios; the results are presented in Table 7. This approach can relax the tight structure of the SBM25 portfolios, as advocated by Lewellen et al. (2010).

Table 7.

25 Size and book-to-market portfolios with five industry portfolios: OLS, GLS, and GMM results.

This table summarizes the test statistics of model diagnostics from Fama–MacBeth’s two-pass regressions and GMM for the SBM25 portfolios and five Fama-French 5 portfolios. FF5 refers to Fama and French’s (2015a) five factor model. Panel A summarizes the OLS estimates, Panel B summarizes the statistics of GLS estimates and Panel C summarizes the GMM estimations. p(R² = 1) and p(R² = 0) are p-value of R² in testing R² = 1 and R2 = 0, respectively. p(F-test) is the p-value of F-test. Following Kan et al. (2013), we report the OLS or GLS cross-sectional R² (in %) and corresponding p-value. &R² is an alternative OLS R², following Campbell and Vuolteenaho (2004). The p-value of F-test is the p-value of a generalized version of Shanken’s (1985) CSRT, following Kan et al. (2013). The J-test is Hansen’s (1982) test of the model over-identifying restrictions. The HJ-distance test is the Hansen–Jagannathan distance measure (Hansen and Jagannathan, 1997) whose p-values are obtained from 10,000 simulations as in Jagannathan and Wang (1996). The sample period is from January 1972 to September 2014 for Michel’s (2009) ICAPM and its inflation-augmented model, while the sample period is from July 1963 to September 2014 for other models.

		Michel’s ICAPM	Michel’s ICAPM + inf	Hahn and Lee’s ICAPM	Hahn and Lee’s ICAPM + inf	HKK	HKK model + inf	FF5
Panel A: OLS estimation of models
OLS	R ²	0.94	0.96	0.96	0.97	0.94	0.97	0.97
	&R²	0.23	0.46	0.47	0.67	0.28	0.57	0.58
	p(R² = 1)	0.02	0.15	0.22	0.82	0.00	0.30	0.00
	p(R² = 0)	0.00	0.00	0.00	0.00	0.00	0.00	0.00
	p(F-test)	0.00	0.00	0.06	0.24	0.00	0.04	0.00
Panel B: GLS estimation of models
GLS	R ²	0.10	0.33	0.19	0.36	0.23	0.45	0.30
	p(R² = 1)	0.00	0.02	0.00	0.08	0.00	0.14	0.00
	p(R² = 0)	0.00	0.00	0.03	0.00	0.00	0.00	0.00
	p(F-test)	0.00	0.00	0.00	0.00	0.00	0.02	0.00
Panel C: GMM estimation of models
GMM	HJ-dist	0.40	0.36	0.37	0.34	0.37	0.34	0.34
	p-Value of HJ-dist	0.00	0.03	0.00	0.05	0.00	0.03	0.00
	p-Value of J-test	0.00	0.00	0.00	0.00	0.00	0.00	0.00

OLS: ordinary least squares; ICAPM: intertemporal capital asset pricing model; GLS: generalized least squares; GMM: generalized method of moments; SBM25: 25 size and book-to-market.

The coefficient estimates of factors are similar and consistent with the previous findings. Specifically, the coefficient estimate of inflation remains significant. However, the overall performance of our ICAPMs augmented with inflation demonstrates somewhat weaker performance. For example, our models can no longer pass the p-value test of the J-test statistic and Shanken’s F-statistic of GLS, as it is well-known that industry portfolios are much harder to predict.

We then compare the performance of the models, although all the models are misspecified for the 30 portfolios. First, we compare the performance of competing models by means of the value of the R² statistic and HJ-distance. Our augmented models have an overall robust and superior performance as described in the following. The R² statistics in the Fama–MacBeth OLS regression in our augmented models are slightly higher than the original ICAPMs. The evidence is clearer in the Fama–MacBeth GLS regression because Michel’s (2009) ICAPM augmented with inflation attains an R² statistic of 0.33, while Michel’s (2009) ICAPM has a low R² statistic of 0.10. As with the Fama–MacBeth GLS estimation of another pair of competing models, the R² statistic of Hahn and Lee’s (2006) ICAPM is only 0.19, while the R² statistic of Hahn and Lee’s (2006) ICAPM augmented with inflation is as high as 0.36. In addition, the R² statistic of the HKK model is 0.23, while the HKK model augmented with inflation achieves a higher R² statistic of 0.45. In the GMM estimations, the inflation factor helps decrease the values of the HJ-distance of each model by approximately 3 percentage points.

Apart from examining the model fit in terms of the values of the R² statistic and HJ-distance, we evaluate the model performance via the p-value approach. Our three inflation-augmented models show better performance in the p-values of the GLS R² statistic (R² = 1), and all marginally pass the HJ-distance test at the 0.01 level of significance, whereas the original ICAPMs and HKK model fail to marginally pass these p-value tests. In Fama–MacBeth’s OLS regression, the p-value of the R² statistic (R² = 1) consistently supports the overall superiority of our augmented models. Moreover, all our augmented models have some significant explanatory power because they all pass the p-value tests of the OLS and GLS R² statistics (R² = 0).

Finally, Fama and French (2015a, 2015b) add the factors of profitability and investment to the FF3 and find that this five-factor model (FF5) outperforms the FF3 in international equities. We therefore take FF5 rather than FF3 as our benchmark model. We compare the performance of our inflation-augmented multi-factor models with Fama and French’s (2015a) FF5 in Table 7. We directly compare the value of the R² statistic and the value of HJ-distance as two key indicators to evaluate the models on the 30 structure-relaxed portfolios. We find that Hahn and Lee’s (2006) ICAPM model augmented with inflation outperforms the FF5 in both OLS and GLS. The HKK model augmented with inflation does a better job in terms of having a higher value of GLS R² and performs as well in terms of the statistics of the OLS R² and HJ-distance. However, Michel’s (2009) ICAPM augmented with inflation appears to show weaker performance than the Fama–French five-factor benchmark model. These findings suggest that the Fama–French five-factor benchmark model does not often outperform our proposed multi-factor models in pricing the cross section of the SBM25 plus five industrial portfolios (a total of 30 portfolios), although it performs well in investment- or profitability-sorted portfolios.

We also augment the FF5 model with inflation in Table 8 of Appendix 1. We find that inflation is also significantly priced in the “FF5 + inf” model, in which the “FF5 + inf” model means that the FF5 is augmented with inflation. The “FF5 + inf” model performs better than FF5 in the OLS R² and GLS R² and the corresponding p-value of R² and Shanken’s F-statistic. Therefore, we conclude that inflation can significantly provide some additional information beyond the FF5 in pricing the cross section of SMB25.

We also follow one of Lewellen et al.’s (2010) suggestions to test a few of other portfolios. Turn to Table 9 of Appendix 1, we also find augmented inflation to the original model can successfully improve the performance of the original Michel’s ICAPM, Hahn and Lee’s ICAPM, and HKK model particularly for 25 size and operating profitability portfolios. We also use p-value approach for robustness. Overall, we can confirm superiority of our inflation-augmented models via p-value approach. Note that Fama and French (2015a) suggest testing 25 size and operating profitability portfolios and they document the importance of operating profitability anomaly, where the operating profitability anomaly cannot be explained by CAPM satisfactorily. Furthermore, we also find inflation can help the original model to price 25 size and net share issue portfolios using cross-sectional tests.

5. Concluding remarks

We augment the inflation factor into two recent ICAPMs and the HKK model: Michel’s (2009) ICAPM, Hahn and Lee’s (2006) ICAPM, and the HKK model. The two original ICAPM models and the HKK model fail to adequately explain the return variations across the SBM25 portfolios in terms of the p-values of the R² statistic (R² = 1) and other test statistics for the model diagnostics. We find that inflation is a key missing factor for Michel’s (2009), Hahn and Lee’s (2006), and the HKK models. Furthermore, our inflation-augmented models not only achieve a remarkable model fit but also alleviate the over-fitting problem.

Footnotes

Appendix 1

Table 9.

25 Size and operating profitability portfolios: OLS, GLS, and GMM results.

This table summarizes the test statistics of model diagnostics from Fama–MacBeth two-pass regressions and GMM for the 25 size and operating profitability portfolios. Panel A summarizes the OLS estimates, Panel B summarizes the statistics of GLS estimates and Panel C summarizes the GMM estimations. p(R² = 1) and p(R² = 0) are p-value of R² in testing R² = 1 and R² = 0, respectively. p(F-test) is the p-value of F-test. Following Kan et al. (2013), we report the OLS or GLS cross-sectional R² (in %) and corresponding p-value. &R² is an alternative OLS R², following Campbell and Vuolteenaho (2004). The p-value of F-test is the p-value of a generalized version of Shanken’s (1985) CSRT, following Kan et al. (2013). The J-test is Hansen’s (1982) test of the model over-identifying restrictions. The HJ-distance test is the Hansen–Jagannathan distance measure (Hansen and Jagannathan, 1997) whose p-values are obtained from 10,000 simulations as in Jagannathan and Wang (1996). The sample period is from January 1972 to September 2014 for Michel’s (2009) ICAPM and its inflation-augmented model, while the sample period is from July 1963 to September 2014 for other models.

		Michel’s ICAPM	Michel’s ICAPM + inf	Hahn and Lee’s ICAPM	Hahn and Lee’s ICAPM + inf	HKK	HKK model + inf
Panel A: OLS estimation of models
OLS	R ²	0.94	0.98	0.94	0.98	0.97	0.99
	&R²	0.08	0.66	0.035	0.66	0.62	0.84
	p(R² = 1)	0.00	0.43	0.00	0.60	0.13	0.61
	p(R² = 0)	0.00	0.00	0.00	0.00	0.00	0.00
	p(F-test)	0.02	0.30	0.00	0.83	0.06	0.44
Panel B: GLS estimation of models
GLS	R ²	0.19	0.37	0.22	0.43	0.26	0.49
	p(R² = 1)	0.00	0.13	0.00	0.27	0.01	0.37
	p(R² = 0)	0.00	0.00	0.00	0.00	0.00	0.00
	p(F-test)	0.00	0.09	0.00	0.22	0.02	0.22
Panel C: GMM estimation of models
GMM	HJ-dist.	0.28	0.26	0.27	0.24	0.27	0.23
	p-Value of HJ-dist.	0.00	0.12	0.00	0.20	0.00	0.31
	p-Value of J-test	0.02	0.06	0.00	0.09	0.00	0.17

OLS: ordinary least squares; GMM: generalized method of moments; GLS: generalized least squares; ICAPM: intertemporal capital asset pricing models.

Final transcript accepted 1 December 2016 by Millicent Chang (AE Finance).

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

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