*4.4. Robustness Test*

As discussed earlier in this paper, our sample period includes the Global financial crises (2007–2009). Therefore, we break the sample into three sub-periods based on a combination of global and domestic market conditions, to confirm that our findings are robust. From January 2002 to December 2006 (pre-crises), from January 2007 to December 2010 (crises period) and from January 2011 to December 2015 (post-crises). Table A2 (Panel A, Panel B, and Panel C) in the appendix examines the time varying behavior of the three-factor model and size and book-to-market factors.

The *R*<sup>2</sup> of the six size-B/M portfolios' regressions in the post-crises period, with an average of approximately 81.82%, is higher than the other sub-periods, followed by crises (79.12%) and pre-crises (65.1%) periods, respectively. In the first sub-period (pre-crises), two out of six portfolios have statistically significant intercepts, whereas, in the second sub-period (crises), one portfolio has significant intercept, and in the third sub-period (post-crises), all the portfolios have insignificant intercepts. However, the magnitude of the intercepts is very nominal, ranging from 0.0001 to 0.0153 in the first sub-period, from −0.0143 to −0.0014 in the second sub-period, and from −0.0038 to 0.0039 in the third sub-period. All the coefficients on market factor across all the three sub-periods are significant at 1% level. With regard to size factor, the six size-B/M portfolios across the sub-periods exhibit varying degrees of sensitivity to the size factor, SMB. However, generally, the small portfolios have a large and positive sensitivity to SMB, whereas big portfolios have a small and negative sensitivity in first two sub-periods, and a nominal but positive sensitivity in the last sub-period. The average coefficients on small portfolios in each sub-period; pre-crises, crises and post-crises, are comparatively higher (0.762, 0.933 and 1.788, respectively) than the average coefficients on big portfolios (0.238, −0.067 and 0.288, respectively).

Our results show that size premium is getting stronger over the time period in terms of both the coefficients and the significance. Finally, value factor across the three sub-periods; pre-crises, crises and post-crises, high B/M ratio portfolios have a positive and large sensitivity to HML (0.908, 0.267 and 0.780, respectively), while low B/M stocks have a small and negative sensitivity to HML ( −0.092, −0.733 and −0.220, respectively). The significance of the value factor is the highest in the post-crises period with approximately similar magnitude as in the pre-crises period. Our results for value factor confirm the findings of Davis et al. (2000), where significance of value premium increased from (*t* = 2.8) to (*t* = 3.38) in the recent sub-period. The value premium in our study is getting stronger over the time period. This is in contrast to the finding of Chung et al. (2016), where it was concluded that value premium is getting weaker over the time period in the Australian market.

It has been discussed that the data from the low-risk state are consistent with CAPM, whereas data from the high-risk state are inconsistent with CAPM (Huang 2000). The sample was divided into two different regimes (low risk and high risk). The results suggested that the data from the high risk regime violate CAPM. However, the data from the low risk state are consistent with CAPM. First, we regressed all the stocks on the market beta and classified them into 6 risk-based portfolios (B1 (high), B2, B3, B4, B5 and B6 (low)) to measure whether size and value premiums exist in all risk regimes. B1 represents stocks of the highest risk category (market beta), whereas B6 represents stocks at the lowest risk level. All portfolios contain an equal number of stocks.

Table 10 represents the estimation results of the three-factor model within a time-series context for each of the six risk-based portfolios. The regression results show that the coefficients of the market, size and value premiums are positive and significant at 1% level. The intercept is statistically insignificant for all the six portfolios and the *R*<sup>2</sup> values ranges between 36.56% and 74.68%. Our results support the existence and significance of the size and value premiums across all the risk profiles (regimes). However, the medium-ranked portfolios (B2, B3 and B4) have a higher explanatory power, and a higher level of significance for loadings on SMB and HML, with somewhat similar magnitude.


**Table 10.** Three factor regression on monthly excess returns of portfolios formed on risk profile (market beta).

Note: Author's calculation. The table reports the estimation results of the three-factor model. Stocks are sorted into six risk-based portfolios. B1 contains securities of the highest risk level (highest market beta) whereas B6 contains the lowest risky securities. *t*-stats are in parenthesis, \*\* and \* indicate significance at 5% and 1% level, respectively. The sample period is 2002:01–2015:12 (168 monthly observations). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/) and the official website of the State Bank of Pakistan (http://sbp.org.pk/).

Petkova (2006) noticed a moderate explanatory power of the Fama–French factors on stock returns in the presence of macroeconomic risk factors. Elgammal et al. (2016) investigated the relationship between default premium and size and value premiums in the US market. They suggested that the default premium has explanatory power for value and size premiums. Baek and Bilson (2015) confirmed the existence of size and value premiums in both financial and nonfinancial firms. Additionally, they clarified that the financial firms are also sensitive to interest rate risk premium. Since our sample also includes financial firms, we examine the augmented four-factor model by including term structure premium into the Fama French three-factor model. The term structure premium (TSP) is calculated by finding the difference between the cut-off yield on ten-year Pakistan investment bonds (PIBs) and three-month Pakistani Treasury bill rate. By introducing TSP into the model, the relationship between excess returns and risk factors is modelled as:

$$E(R\_i) - R\_f = a\_i + \beta\_i \left[ E(R\_{\text{ill}}) - R\_f \right] + s\_i(SMB) + h\_i(HML) + ts\_i(TSP) + \varepsilon\_i \tag{5}$$

where, *E*(*Ri*) − *Rf* is the portfolio *i's* return in excess of risk-free rate *Rf* , *αi* is the intercept of the regression equation representing the non-market return component, *E*(*Rm*) − *Rf* is the market risk premium (market portfolio return in excess of risk-free rate), SMB (small minus big) is the return on small size stocks minus return on big size stocks captures size premium, HML (high minus low) incorporates value premium that is the difference between returns of value stocks (high B/M ratio) and growth stocks (low B/M ratio), and *TSP* (term structure premium) is calculated by finding the difference between the cut-off yield on ten-year PIBs and three-month T-bills of Pakistan. *β<sup>i</sup>*, *si*, *hi*, and *tsi* are the slopes of expected risk premium of portfolio *i* to the market factor, size factor, value factors and term structure premium in the regression, respectively, while *i* represents the random return component due to unexpected events related to a particular portfolio.

Results reported in Table 11 show that there is a negligible increase in the average adjusted *R*<sup>2</sup> due to the addition of TSP (from 71.23% to 71.73%). In contrast, there is a huge increase in the significance of the average intercept (from one portfolio to three portfolios) and magnitude of the average intercept (from 0.004 to 0.008). Our results demonstrate that SMB, HML and market factors remain robust to the inclusion of the term structure premium. Our findings for the three-factor model are robust across various portfolio construction techniques.


**Table 11.** Augmented four-factor regression on monthly excess returns of portfolios formed on size and B/M ratio (variable basket).

Note: Author's calculation. The table reports the estimation results of an augmented Fama-French four-factor model that includes term structure premium. Stocks are sorted into six size-B/M portfolios (SL, SM, SH, BL, BM and BH). Term structure premium is calculated by finding the difference between the cut-off yield on ten-year PIBs and three-month T-bill of Pakistan. *t*-stats are in parenthesis, \*\*\*, \*\* and \* indicate significance at 10%, 5% and 1% level, respectively. The sample period is 2002:01–2015:12 (168 monthly observations). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/) and the official website of the State Bank of Pakistan (http://sbp.org.pk/).

#### **5. Predictive Ability of the Three Factors for Future Economic Growth**

Fama and French (1992, 1993, 1996, 1998) argue that SMB and HML act as state variables that predict future variations in the investment opportunities established in the context of intertemporal capital asset pricing model (Merton 1973). Liew and Vassalou (2000) attempt to link the return-based factors with future growth in the macro-economy. They conclude that HML and SMB contain significant information about future GDP growth, and risk-based explanation for the returns of SMB and HML is plausible. The evidence will enhance our understanding of whether or not these factors relate to underlying economic risk factors.

In this section, we discuss the third main objective of the study. This objective is to examine the relevance of the market factor, SMB, and HML with future GDP growth of Pakistan using univariate and multivariate regression analysis. Along with these factors, we include Treasury bill rate and term structure premium to predict Pakistan's GDP growth one year ahead. That is, we explore the ability of these factors at year Yt−<sup>1</sup> to forecast the GDP growth for year Yt. The annual GDP data is obtained from the official website of the Asian development bank, whereas the data for T-bills and PIBs are obtained from the official website of the State Bank of Pakistan. Term structure premium is calculated by finding the difference between the cut-off yield on ten-year Pakistan investment bonds and three-month Pakistani Treasury bill rate, while GDP growth is calculated as the continuously compounded growth rate in Pakistan's gross domestic product. To obtain the yearly values, we have calculated the average (mean) of the monthly market risk premium (MKT), SMB, HML, Treasury bill rate (TB) and term structure premium (TSP) within each year (12 months).<sup>8</sup> The following equation represents the model:

$$GDP\_{\mathcal{S},t} = a + \beta \left[ E(R\_{m,t-1}) - R\_{f,t-1} \right] + sSMB\_{t-1} + hHML\_{t-1} + fBCV\_{t-1} + \epsilon\_{t} \tag{6}$$

<sup>8</sup> See Boamah (2015) and Liew and Vassalou (2000) for an extensive overview of the methodology.

where, GDPg,t represents the GDP growth at time *t* and BCVt−<sup>1</sup> represents the business cycle variables, which refers to the term structure premium and the three-month Treasury bill rate.

Before we proceed with the main test, we examine the stationarity of the variables. The market factor and HML are stationary, while SMB, GDP growth, T-bill and term structure premium are taken as first-order difference. The absence of a unit root in the series of returns is confirmed by augmented Dickey-Fuller test and Phillips-Perron test, with trend and intercept. It is carefully observed that all the variables are stationary at the time we perform regressions. Next, we examine various versions of Equation (6) and present the results in Table 12. To check whether the remaining residuals are independent and identically distributed (i.i.d.), we have conducted the BDS test by Broock et al. (1996) and no nonlinearity is found.

The evidence shows that in univariate regressions, the market factor and size premium show significant association with future growth of the Pakistani GDP, whereas value premium, Treasury bill and term structure premium indicate an insignificant relationship with future growth in GDP.

It is further evident that only the coefficient of the market factor is positive, whereas the coefficients of SMB, HML, TB and TSP are negative. The explanatory power of the univariate regressions is 63.03%, 32.79%, 1.30%, 9.32% and 1.02% for the market, SMB, HML, TB and TSP, respectively. The findings indicate that the predictive ability of the market factor and SMB for the growth of the Pakistan's GDP is non-trivial.

In a two-factor model consisting of market factor and SMB, HML, TB and TSP, the result indicates that the coefficients of market factor are all positive and statistically significant. The loadings on HML, SMB, Treasury bill and term structure are negative, but insignificant. The *R*<sup>2</sup> values are relatively higher in the two-factor regression, ranging from 63.32% (market and TSP) to 68.86% (market and TB). The two-factor model consisting of SMB and HML indicates that including HML into the regression model does not subsume the significance of SMB factor. The negative coefficients of SMB are similar to the findings of Liew and Vassalou (2000) for Switzerland and Japan.


**Table 12.** The information content of market, SMB and HML for future economic growth.

Note: Author's calculation. The table reports the estimation results of the Fama-French three-factor model to predict future GDP growth of Pakistan. The MKT, SMB, HML and BC are correspondingly the excess return to the market risk premium, size premium, value premium, and business cycle variables (Treasury bill (TB) and term structure premium (TSP)). *t*-stats are in parenthesis, \*\*\*, \*\* and \* are the 10%, 5% and 1% significance level, respectively. The sample period is 2002–2016 (168 monthly observations of risk factors (converted into annual values), and 14 annual observations of GDP growth). Source: the official website of the Pakistan stock exchange (https://www.psx.com.pk/), the official website of the State Bank of Pakistan (http://sbp.org.pk/) and the Asian development bank (https://www.adb.org/data/south-asia-economy).

In a multivariate regression analysis, Table 12 further reports that including Treasury bill, term structure premium or both in the model does not subsume the relevance of the SMB. The market factor has the strongest relevance with the GDP and deteriorates all the other factors. However, inclusion of HML and business cycle variables (TB and TSP) does not eliminate the forecasting of SMB. The evidence in this study suggests that the market factor and SMB possess the information content for one year ahead Pakistan's GDP growth. The negative relation of SMB with future economic growth, presumably, indicates that the investors would rather hold the big capitalization stocks when they notice that the economy is in bad state (low or instable growth).
