7. Hypothesis tests of joint significance of the regressions' alphas and regressions

Having presented the methodology and statistical results, in this part, I present an answer to important question if the estimated models can completely capture expected returns. To obtain a more absolute answer to this question GRS f-tests were conducted on results obtained from the first hypothesis' portfolio regressions. The GRS statistic is used to test if the alpha values from regressions are jointly indistinguishable from zero.

If a model completely captures expected returns, the intercept should be indistinguishable from zero. Hence, the first hypotheses are:

H0: α^1 = α^2 = … = α^16 = 0 (the regression alpha is not significantly different from zero).

H1: α^1, α^2, …, α^16 6¼ 0 (the regression alpha is significantly different from zero).

The second hypotheses for all equations are:

excess return of 1.55%. One explanation could be that even if the biggest-sized companies have low profitability, if it is an aggressive investing company and the expectation of the market participants is positive, then a monthly average return of 1.55% would be justified. As you may see in Panel C of Table 3, big companies with aggressive investing have the highest returns.

Factor spanning regressions are a means to test if an explanatory factor can be explained by a combination of other explanatory factors. Spanning tests are performed by regressing returns of one factor against the returns of all other factors and analysing the intercepts from that regression. Table 4 shows regressions for the 5F-FF model's explanatory variables, where four factors explain returns on the fifth. In the RM-Rf regressions, the intercept is not statistically significant (t = 0.55). Regressions to explain HML, RMW and CMA factors are strongly positive. However, regressions to explain SMB show insignificant intercept, with intercept of 0.27% (t = 1.34). These results suggest that removing either the RM-Rf or SMB factor would not hurt the mean-variance-efficient tangency portfolio produced by combining the remaining four

Intercept RM-Rf SMB HML RMW CMA R2

Coefficient 0.45 0.34 0.41 0.29 0.21 0.20

Coefficient 0.27 0.02 0.00 0.07 0.01 0.12

Coefficient 0.55 0.04 0.01 0.06 0.10 0.19

Coefficient 1.06 0.03 0.12 0.06 0.07 0.17

Coefficient 1.04 0.02 0.03 0.11 0.07 0.16

RM-Rf is the equal-weighted return on BIST-100 Index minus the 3-month Tryribor rate. SMB is the size factor, HML is the value factor, RMW is the profitability factor, and CMA is the investment factor. The factors are constructed using individual sorts of stocks into two size groups and three B/M groups, three OP groups or three Inv groups. Bolded and

Table 4. Factor spanning regressions on five factors: spanning regressions using four factors to explain average returns

t-Stat 0.55 1.17 1.81 1.33 1.02

t-Stat 1.34 1.17 0.07 1.32 0.29

t-Stat 2.19 1.81 0.07 0.82 1.50

t-Stat 4.18 1.33 1.32 0.82 1.01

t-Stat 3.95 1.02 0.29 1.50 1.01

shaded t-statistics indicate the significance at the 5% level.

on the fifth (June 2000–May 2017, 204 months).

6. Factor spanning regressions

78 Financial Management from an Emerging Market Perspective

factors.

RM-Rf

SMB

HML

RMW

CMA

H0: α^1 = α^2 = … = α^48 = 0 (the regression alphas are jointly indistinguishable from zero).

H1: α^1, α^2,…, α^48 6¼ 0 (the regression alphas are jointly distinguishable from zero).

Finally, average individual regression alphas and joint GRS regression f-values were used together in order to compare the performance between the tested models.

#### 7.1. The GRS regression equation

The GRS test was developed by Gibbons et al. [24] and serves as a test of mean-variance efficiency between a left-hand-side collection of assets or portfolios and a right-hand-side model or portfolio. The following regression defines the GRS test:

$$f\text{GRS} = \frac{T}{N} \times \frac{T-N-L}{T-L-1} \times \frac{\widehat{\alpha}' \times \widehat{\Sigma}^{-1} \times \widehat{\alpha}}{1 + \overline{\mu}' \times \widehat{\Omega}^{-1} \times \overline{\mu}} \sim F(N, T-N-L) \tag{15}$$

where α^ is a N�1 vector of estimated intercepts, Σb an unbiased estimate of the residual covariance matrix, μ a L�1 vector of the factor portfolios' sample means and Ωb an unbiased estimate of the factor portfolios' covariance matrix.

The GRS test is used in this study to determine whether the alpha values from individual model regressions are jointly non-significant and hence to find out if a model completely captures the sample return variation. As intercepts from individual regressions approach zero, the GRS statistic will also approach zero. However, since the GRS statistic derives its results from comparing the optimal LHS and RHS portfolios, the resulting statistic is not strictly comparable between models.

#### 7.2. Model performance

A set of several summary metrics were deployed in order to compare the performance of the asset pricing models. GRS statistics and average alpha values were used as the main two statistics in order to determine how good the different asset pricing models performed in explaining portfolio returns. In addition to these statistics, average absolute alpha spread was added for a more complete picture of the alpha results. Furthermore, different models' explanatory power was measured using R<sup>2</sup> values.

If a capital asset pricing model (CAPM, three-factor model or five-factor model) completely captures expected returns, the intercept (alphas) is indistinguishable from zero in a regression of an asset's excess returns on the model's factor returns.

Table 5 shows the GRS statistics of [24] that tests this hypothesis for combinations of LHS portfolios and factors. The GRS test easily rejects all models considered for all LHS portfolios and RHS factors. The probability, or p-value, of getting a GRS statistic larger than the one observed if the true intercepts are all zero is shown in column 'pGRS'. One can see from Table 5 that except CAPM in Panel A and Panel C, sets of left-hand-side returns, the p-values round to zero to at least three decimals. Only five-factor model in Panel C has a p-value of 0.30, and it is still significant at the 5% level.


The table tests the ability of CAPM, 3F-FF and 5F-FF models to explain monthly excess returns on 16 size-B/M portfolios (Panel A), 16 size-OP portfolios (Panel B), 16 size-Inv portfolios (Panel C) and a joint sample of all 48 portfolios (Panel D). For each panel, the table shows the tested model; the GRS statistic testing whether the expected values of all 16 or 48 intercept estimates are zero; the p-value for the GRS statistic; the average absolute value of the intercepts, Avg |α|; the average absolute deviation from the average intercept; and the average R2 . Bolded and shaded GRS statistics indicate the significance at the 5% level.

Table 5. Summary of statistics for model comparison tests: summary of statistics for tests of CAPM, 3F-FF and 5F-FF models (June 2000–May 2017, 204 months).

Fama and French [19] state that they want to identify the model that is the best (but imperfect) story for average returns on portfolios formed in different ways, and they use absolute value of average alphas (|α|) and two more statistics (for a detailed description of these statistics, see ([21], pp. 10). Following [22], in this study I use only average absolute values of alphas (|α|) and the average absolute deviation from the average intercept (Avg j j α � α ). Both [19, 22] pay specific importance of the absolute values of alphas. Relatively, small values of Avg|α| or Avg j j α � α in equations are regarded as better in identifying the best model. However, as is seen in Table 5, Avg|α| of CAPM model in Panel A has the lowest value, but fGRS and pGRS show that we cannot reject the hypothesis that the regression alpha is significantly different from zero. The same case is valid for the CAPM in Panel C in Table 5. Therefore, I think concentrating on the magnitude of the alpha values or absolute deviation from the average intercept (Avg j j α � α ) in the equations to decide on the best performing model may not be a healthy approach. Keeping this in mind, my main conclusion from the model comparison tests follows.

The GRS test in Panel D clearly rejects the second hypothesis for all models tested. Furthermore, the GRS test in Panel D suggests that the 5F-FF model is the model closest to a complete description of asset returns. Beware that, as discussed in [19], fGRS values between models cannot be strictly compared. Instead, the fGRS is mainly interpreted as a test to reject or accept a model's explanation of returns on a set of left-hand-side portfolios. The fGRS in [21, 22] is used in comparisons between models only in combination with a comparison of average alpha values. Fama and French [19] instead use the numerator of a GRS regression as a comparison value when choosing which factors to include in a model. In every panel of Table 5, explanatory power measured by average R2 clearly improves with the inclusion of more factors. For this reason, metrics for explanatory power are important in providing additional help to compare these types of asset pricing models.

The main conclusions from the performance comparison tests can be summarised as follows. A GRS test on the joint set of all tested portfolios clearly rejects all tested models as complete descriptions of average returns. The CAPM model elicits the lowest average absolute alpha values of the three tested models throughout all tests but shows a statistically insignificant fGRS value compared to other models. Considering all the evidence in Table 5, it is clear that the 5F-FF model shows the strongest performance out of the three models for the sample. In the following section, I provide alpha values from individual left-hand-side portfolio regressions, and I re-examine the alpha values from a different perspective.
