The small-firm effect refers to the empirical observation that small-capitalisation stocks tend to outperform on average larger capitalisation stocks over the long run and that this outperformance cannot be attributed solely to differences in market risk (e.g., Banz, 1981; Reinganum, 1981; Hawawini and Keim, 1999). Several papers have challenged this result (e.g., Brown et al., 1983; Bhardwaj and Brooks, 1993; Dimson and Marsh, 1999; Horowitz et al., 2000; Schwert, 2003). More recently, Van Dijk (2011) noted that while the small-cap premium disappeared through the latter two decades of the twentieth century, it has made a strong resurgence since the beginning of the millennium: Indeed, US small-caps returns earned a premium over large caps of 6% per year over the period 2000–2014.

This paper provides a new approach to assessing the time-varying nature of the small-cap premium for Canada and the US, by testing the hypothesis that the link between macroeconomic and financial variables that drives this premium (or discount) is dependent on the business cycle. The hypothesis is motivated by the empirical regularity that small-cap firms outperform (underperform) large-cap firms during periods of economic expansion (contraction). Only a few studies have looked at this issue to date (e.g., Chan et al., 1985; Switzer, 2010, 2013). Figure 1 provides a graphical illustration of this phenomenon for the US experience since 1926. Comparable data are available for Canada since 1970 and are projected on Figure 2. In both cases, a downward trend in the small-cap premium can be seen at the onset of recessionary periods, including the most recent short Canadian recession, which lasted from November 2008 to May 2009, and the US Great Recession from May 2007 to June 2009, denoted as the shaded areas within the graphs.

Over the period 1926–2014 for the US, during expansions, the average annualized premium is a sizable 5.44%, while during recessions, there is a small-cap discount of 6.23%. Small-cap stocks may be more exposed to systematic default risk related to the business cycle. This study employs the smooth transition autoregressive model (STAR model) as the approach for estimating the conditional asset pricing dynamic and, hence, to explore the relationship between the small-firm premium and financial and macroeconomic variables in the Canadian and US economies.

Two of the most important asset pricing models in modern finance theory are the capital asset pricing model (CAPM) proposed by Sharpe (1964) and Lintner (1965) and the arbitrage pricing theory (APT) suggested by Ross (1976). Both specifications assert that the expected return of an asset is a linear function of risk; in CAPM, risk is measured by the covariance of the asset return with the return to the market portfolio, and in APT, risk is estimated as exposure (covariance) to a set of factors, which may comprise the market portfolio, amid others. Although the CAPM and the APT specifications suggest that the relationship between return and risk is linear, there is increasing empirical evidence suggesting that applications of these factor models should control for time-varying risk factor premiums and risk factor exposures or betas, and that doing so provides some improvement. For instance, Blume (1971), Levy (1971), Fabozzi and Francis (1978), Chen (1981), Ferson and Harvey (1991, 1999) and Ferson and Korajczyk (1995) showed that estimated beta coefficients depend on whether the capital markets are under high or low volatility regimes, in linear versions of the CAPM. These empirical observations could explain the reason the predictable power of several factors employed in the literature to predict excess aggregate equity returns, like dividend yields, term spreads and default spreads, decreased or even disappeared over the nineties, as documented by Goyal and Welch (2008) and Ang and Bekaert (2007).

Time-varying betas and factor loadings can be justified by the reasonable supposition that the relative risk and the expected excess returns of securities are related to their sensitivity to changes in the state of the economy. In other words, the estimated beta coefficients under a low volatility regime differ significantly from their counterparts under a high volatility regime. Several studies showed superior performance of dynamic conditional versions of the CAPM and APT asset pricing models in explaining and predicting stock returns relative to their static and unconditional forms (counterparts) (see, e.g., Bansal and Visnawathan, 1993; Jagannathan and Wang, 1996). Ammann and Verhofen (2008) documented that time variation of betas in the CAPM and the time variation of the coefficients for the size factor (SMB) and the distress factor (HML) in the three-factor model enhance the empirical performance in describing the cross-section of returns of the S&P500 constituents.1 Goyal and Welch (2008) and Henkel et al. (2011) showed that the predictability of conditional models appears stronger during periods of economic contractions, as opposed to periods of expansion.

This article extends the Goyal and Welch (2008) and Henkel et al. (2011) papers which examine the performance of variables that have been suggested by the academic literature to be good predictors of the equity premiums, by investigating the size premium rather than instead of the aggregate market risk premium. Moreover, this paper applies a different approach to those of Perez-Quiros and Timmermann (2000), who modelled excess returns of size-sorted decile portfolios by investigating the small-cap premium measured as the difference between returns of the 33% smallest firms and 33% largest firms. We found that the significances of some risk exposures coefficients (i.e., inflation and term risk factors) appear negligible during business cycle expansions but nontrivial during contractions.

As mentioned previously, the model estimated is the smooth transition autoregressive (STAR) econometric model, which shows that the small-cap premium is related to some macroeconomic and financial variables among the default, term, inflation, short-term rate and dividend yield factors, and that relationship is driven by the economic cycle in the United States as well as Canada.

The organization of the paper is as follows. Section 2 reviews the literature that examines the cyclical nature of the small-cap premium. Section 3 discusses the modelling approach. Section 4 provides a description of the data. The empirical results follow in Section 5. The paper concludes with a brief summary in Section 6.

Several studies have examined the time-varying nature of the small-cap premium. Some of these studies link the premium to conventional macroeconomic risk factors, using an efficient markets perspective. Others adopt a behavioural approach. Both approaches, which are presented below, converge in the nature of their predictions: that the premium should reflect the state of the economy over the business cycle.

One of the most well-known explanations of the small-firm premium is that firm size is a proxy for some undiversifiable and undisclosed macroeconomic or other specific risk factor(s) that influence(s) the dynamic of expected asset returns. If so, the observed small-firm premium reflects investors’ compensation for the exposure to risk. This risk-based approach is even more discernible and apparent, according to numerous studies mentioned below, by the fact that during expansionary periods, small firms usually grow faster than large, mature firms. In contractionary periods, on the other hand, small firms are more exposed to bankruptcy risk and perform relatively poorly. To reduce exposure to such risks, investors are more prone to exit such stocks in search of safer investments.

Fama and French (1992, 1993, 1996, 2015) are the leading proponents of this risk-based approach.2 In their most recent study (Fama and French, 2015), they introduced a five-factor model that includes size (SMB), value, profitability and investment patterns to explain average stock returns. While their model has been shown to perform better than the three-factor model of Fama and French (1993), it is unable to capture the returns of small-cap stocks, whose returns are similar to those of firms with considerable investment activity, despite low levels of profitability. Their study does not develop a comprehensive theory for what the underlying systematic risk factors in their model represent. Nor do they look specifically at business cycle effects, however. It may be the case that the outperformance of small firms during expansionary periods may be due to their reaping the rewards from staying the course on their investments during recessions.

Perez-Quiros and Timmermann (2000), in a paper closely related to ours, used a Markov switching model with Center for Research in Securities Prices (CRSP) data over the sample period January 1954–December 1997. They modelled excess returns on each of the size-sorted decile portfolios as a function of an intercept term and lagged values of the one-month T-bill rate, a default premium, changes in the money stock and the dividend yield. They demonstrated that small firms display the highest degree of asymmetry in their conditional return distribution across recession and expansion states. In a recession, small firms’ risk is most strongly affected during recessions by worsening credit market conditions as reflected in higher default premia and higher interest rates. They did not account for the possible differential effects of inflation or the term structure of interest rates, however. Our paper revisits this question using more recent data and a more flexible modelling approach than Perez-Quiros and Timmermann (2000), i.e., the smooth transition autoregressive model (STAR).

Liew and Vassalou (2000) investigated whether the dynamics of the Fama–French factors, as well as the Carhart (1997) momentum factor, can be attributed to the future GDP growth across ten countries. Analyzing ten countries between 1978 and 1996, they showed that the SMB factor carries significant information about future GDP growth that is not related to the information included in the market factor. Zhang et al. (2009) also discovered a positive link between the SMB factor and future GDP growth and a negative relationship between SMB and unexpected inflation. Small-cap stocks were also found to generate greater returns than large-cap stocks when the short-term rates are low, and the term spread is high. Petkova (2006) asserted that the SMB portfolios may be linked to shocks in state variables that forecast excess market returns. A model that relates average stock returns to variations in aggregate dividend yield, default spread and short-term Treasury-bill rates was shown to outperform the Fama and French (1995) three factor model in explaining the cross-sectional variation in equity returns. When loadings on the shocks in the predictive variables were incorporated in the model, the loading on the SMB factor lost its explanatory power, implying a strong correlation between the SMB portfolio and the default spread.

Other researchers have tried to explain the small-firm effect by including time variation in the covariance of asset returns with the market return (conditional CAPM) or with consumption growth (conditional consumption CAPM). In these models, the asset’s beta is not constant as in the traditional CAPM but fluctuates to reflect variations in a predefined conditioning variable. The economic reasoning behind a conditional model for the small-firm effect is that small- and large-cap stocks may have different sensitivities to systematic risk in good and bad times. Lettau and Ludvigson (2001) used the log–consumption-to-wealth ratio as a conditioning variable to demonstrate that conditional specifications perform superiorly to unconditional models and comparably to the Fama–French three-factor model in describing the cross-section of average returns. Once the conditioning information conveyed by the log–consumption-to-wealth ratio is incorporated in the specification, no residual small-firm effect persists in the data.

Likewise, Santos and Veronesi (2006) demonstrated that using the fraction of total income produced by wages as a conditioning variable in the CAPM describes the cross-section of 25 Fama–French portfolios formed on value and size. Small-firm stocks display evidence to be riskier in bad times, exhibiting higher betas in market downturns.

Campbell and Vuolteenaho (2004) presented strong empirical evidence that the outperformance of small stocks is caused by higher cash flow risk. The authors rationalised the small-firm effect via an economically induced two-beta model in which they split the CAPM beta of a stock into two elements: one that reflects news about the company’s future cash flows and one that reflects news about the discount rate. Intertemporal asset pricing theory suggests that risk-averse investors avoid cash flow risk (the “bad beta”) more than discount rate risk (the “good beta”). Consequently, the price of cash flow risk is greater than the price of the “good” discount rate risk. In equilibrium, the ratio of the two prices must equal the risk aversion coefficient that makes an investor satisfied to hold the aggregate market. They subsequently showed empirically that small stocks have significantly greater cash flow betas than large stocks, which can explain their higher average returns in the cross-section. The two-beta model hence implies that investors with higher tolerance for risk and a long-term investment horizon will massively invest in these stocks compared to the average investor.

In contrast to (systematic) risk factor explanations within the efficient markets framework of the small-firm premium, other studies have linked the premium to behavioural factors. Chan and Chen (1991) indicated that small firms tend to be firms that have done poorly in the past. The small-firm effect would then be consistent with the overreaction hypothesis of De Bondt and Thaler (1985). Based on Daniel and Titman (1997) and Lakonishok et al. (1992) claimed the small-firm effect may be due to investor preferences for large firms over small firms. This preference might be amplified during economic downturns as investors seek investments with less perceived risk.

More recently, Lemmon and Portniaguina (2006) provided evidence of a negative link between investor sentiment and the dynamic behaviour of returns on small-firm stocks since 1977. They argued that investors are inclined to overvalue small-cap stocks versus large-cap stocks when they are particularly bullish and undervalue them when they are bearish. Sentiment has a particularly great impact on small-cap stock valuations because small firms are usually held by individual traders who are more likely to be influenced by sentiment. However, they uncovered no link between investor sentiment and small-cap returns before 1977 and provided no explanation on the decrease of the small-firm effect after 1981.

In sum, both the efficient markets explanations as well as the behavioural-based approaches suggest that a regime dependent model is appropriate for examining the time-varying nature of the small-cap premium. Furthermore, these approaches imply that the factors affecting the premium are sensitive to the state of the economy. This paper is the first to provide a direct test of these assertions, using recent advances in nonlinear time series modelling for the small-cap premium in the US and Canada.

This section provides a review of the nonlinear econometric model that is employed to investigate the relationship between the small-cap premium and several macroeconomic and financial variables. The STAR model, which was developed by Teräsvirta (1994) and Granger and Teräsvirta (1993), has been extensively used in the past two decades to model nonlinearities in the dynamic properties of many economic time series and summarize and explain the cyclical behaviour of macroeconomic data and business cycle asymmetries.

The smooth transition autoregressive (STAR) model for a univariate time series y t is given by: (1) y t =   α 0 + ∑ i = 1 p α 0 y t − i + F ( ξ t , γ , c ) [ β 0 +   ∑ i = 1 p β i y t − i ] + ε t , where F ( ξ t , γ , c ) is a transition function which controls for the switch from one regime to the other and is bounded between 0 and 1. The scale parameter γ > 0 is the slope coefficient that determines the smoothness of the transition: The higher it is, the more abrupt the change from one extreme regime to the other ξ t . The location or threshold parameter between the two regimes is represented by c , and ξ t is called the transition (threshold) variable, with ξ t = y t − d ( d a delay parameter).

Two standard selections for the transition function are the logistic function (LSTAR) and the exponential function (ESTAR). The LSTAR function is specified as: (2) F =   [ 1 +   exp ( − γ ( ξ t − c ) ) ] − 1 , while the ESTAR function is specified as: (3) F =   1 −   exp ( − γ ( ξ t − c ) 2 ) .

The main difference between these two STAR models relies on how they describe macroeconomic series dynamic behaviour. The LSTAR model reflects the asymmetrical adjustment process that usually characterizes economic cycles: a sharper transition and sharp recovery following business cycle troughs compared to economic peaks. By contrast, the ESTAR specification suggests a symmetrical adjustment dynamic.

To determine the adequate transition function to apply to the data, Teräsvirta (1994) suggested a model selection procedure which is explained and applied in Section 4.

While an exogenous variable could be employed as the transition variable, in this paper, as per the majority of research studies using STAR models, the dependent variable (the macroeconomic proxies) plays this role and d equals one, meaning that the first lagged value of the macroeconomic variable investigated acts at the threshold variable.

In smooth transition autoregression (STAR), all predetermined variables are lags of the dependent variable. An extension to the STAR model is to amend the STAR model to allow for exogenous variables x 1 t , …, x kt as additional regressors. In this study, the applied STAR model includes other exogenous factors, i.e., default, term structure, inflation, short-term interest rate and the dividend yield risk factors. The standard method of estimation of STAR models is nonlinear least squares (NLS), which is equivalent to the quasi-maximum likelihood approach.

Two interpretations of a STAR model are possible. First, the STAR model may be thought of as a regime-switching model that allows for two regimes, associated with the extreme values of the transition function, F( ξ t ; y , c ) = 0 and F( ξ t ; y , c ) = 1, where the transition from one regime to the other is smooth. The regime that occurs at time t is determined by the observable variable   ξ . Second, the STAR model can be said to enable a continuum of states between the two extremes. The key advantage in favour of STAR models is that changes in some economic and financial aggregates are influenced by changes in the behaviour of many diverse agents, and it is highly improbable that all agents respond instantaneously to a given economic signal. For instance, in financial markets, with a considerable number of investors, each switching at different times (probably caused by heterogeneous objectives), a smooth transition or a continuum of states between the extremes seems more realistic.

Macroeconomic variables and premiums related to size, value and liquidity are frequently subject to switching regimes, and this notion of the regime switch implies a sudden abrupt change. Yet, most macroeconomic fundamentals and premiums change regimes in a smooth manner, with transition from one regime to another taking some time.

To reflect this particular characteristic, smooth transition autoregressive (STAR) models have recently been developed and seem to be more appropriate in modelling macroeconomic variables, since smooth transitions between regimes are often more convenient and realistic than abrupt switches.

In contrast to discrete switching models (e.g., Hansen, 1999), for instance, the Markov switching-regime model of Hamilton (1989), smooth transition regression (STAR) models transition as a continuous process dependent on the transition variable which allows for incorporating regime-switching behaviour both when the exact time of the regime change is not known with certainty and when there is a short or long transition period to a new regime which will be reflected by the estimate of the smoothness parameter, which will allow suggesting a very sharp or smooth switch from one regime to the other. Smooth transition models provide additional information on the dynamics of variables and are able to capture regime-switching behaviour in a flexible way and facilitate its economic interpretation (see Van Dijk et al. (2002) for a discussion of the benefits of this class of models). For instance, Sarno et al. (2003) argued that several theoretical models of money demand imply a nonlinear functional form of the aggregate demand for money, characterized by smooth adjustment towards a long run equilibrium.

Hence, this study extends Perez-Quiros and Timmermann’s (2000) paper in three ways: (a) in the methodology to estimate the value of the small-firm premium; (b) by using an extended set of explanations (macroeconomic and financial); and (c) in its approach to approximate economic cycles. Perez-Quiros and Timmermann (2000) used the Markov switching regime to model the small-cap premium, while this paper applies the smooth transition autoregressive (STAR) model to describe its time series dynamic.

This paper investigates the impact of several risk factors that have been assumed to be significant exposures influencing the returns to firms (Chen et al., 1986; Ferson and Harvey, 1991) and that may be independent from the state of the business cycle per se in affecting the return spread between large-cap and small-cap companies. These variables are the default, term, dividend, short-term rate and inflation risk factors. Their description is presented in more details in Appendix A.

Table 2 shows the descriptive statistics of the small-cap premium in the US and in Canada, and the five exposure risk variables i.e., default, inflation, term, dividend yield and the short-term rate (T-bill). On the whole, the small-cap premium is quite high in both the US and Canada. For the US, over the entire sample period, the average premium amounts to 0.0027 per month, or 3.24% per year. The annualized holding period excess return to small caps over large caps is about 2.1% over this period. For the subperiod January 1970–December 2013, the annualized average premium and annualized holding period premium to small caps over large caps increases to 3.60% per year and 2.55% percent, respectively. For Canada, the small-cap premium for the 1970–2013 period is lower than for the US: The annualized average premium is 1.82% a year, while the annualized holding period premium for small caps to large caps amounts to 1.38%. For these comparable periods, 1970–2013, the average Canadian and US risk default premium are higher similar (1.17% vs. 0.39% annualized). The term premium in Canada is also higher (0.0011 vs. 0.0034 monthly, 1.42% vs. 4.15% annualized), as is the inflation differential (0.0034 vs. 0.0030 monthly, 4.15% vs. 3.63% annualized). Finally, the Canadian short-term rate exceeds the US rate (0.0052 vs. 0.0042 monthly, 6.24% vs. 5.04% annualized), while the Canadian and US dividend yield present similar numbers (0.0024 monthly, 2.88% annualized).

Since at least 1981, when Banz (1981) presented empirical evidence that small-cap stocks generated higher average returns than larger firms, a plethora of studies has emerged on the small-firm effect to investigate the validity and persistence of the small-cap premium and to offer explanations for the empirical outperformance of small-cap stocks over the long term. While the small-firm premia in the US and Canada are on average economically sizable over the long term, they also exhibit considerable time-varying properties, which are shown to be linked to the business cycle.

The conventional explanation for the small-cap premium is that it represents a systematic risk premium that needs to be incorporated into practicable asset pricing models. Academics do not suggest that firm size per se is the source of the risk driving the dynamic of expected returns, but that size is a proxy for one or more underlying risk factors associated to smaller firms. Some fairly robust relationships have been uncovered between the small-firm effect and default risk, as well as between the small-firm effect and several macroeconomic variables.

This paper contributes to the extensive literature that has investigated the links between the small-firm premium and various financial and macroeconomic variables using nonlinear regime-switching models. The paper highlights that the US default risk affects the return differential between large-cap and small-cap firms in both Canada and the US and that these relationships are independent of the economic activity phases.

These results are important for policy makers, portfolio managers and academics. The framework for dynamic asset allocation decisions can clearly be enriched by incorporating regimes with specific and, in some cases, independent financial and macroeconomic risk exposures. Forecasting regime shifts that reflect the changing nature of the risk and return opportunity sets will also be invaluable and remains a topic for future work.