A new GARCH model with higher moments for stock return predictability

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Abstract

The main purpose of the paper is to propose a new GARCH-SK predictive regression model that accommodates higher order moments (skewness and kurtosis) in testing the null hypothesis of no predictability. Using an extensive and well-known time-series dataset on stock returns and 19 predictors for the United States, we show that our proposed GARCH-SK model outperforms a model without these higher moments. The superior performance of our proposed model holds both statistically and economically and is robust to different data frequencies.

Introduction

In this paper, we propose a new approach to testing for stock return predictability. This line of research is popular. Its popularity is reflected in the volume of effort devoted to improving existing methods for testing the null hypothesis of no predictability and the volume of empirical applications; see, for instance, Kostakis et al. (2015), and Westerlund and Narayan, 2012, Westerlund et al., 2015. The emphasis on appropriate or improved methods has implications for forecasting stock returns. These forecasts are relevant because they are utilised in devising trading strategies. The effort, therefore, is tailored towards improving our ability to minimise forecasting errors, thus minimising the inherent uncertainty from trading strategy outcomes. There is, in other words, a strong connection between econometric approaches to testing for predictability and its economic significance—a story well-articulated in recent studies (see Narayan and Bannigidadmath, 2015).1

Our contribution is both methodological and empirical in the sense that we propose a model that accommodates higher order moments in testing the null hypothesis of no predictability. Prior literature (see, inter alia, Kraus and Litzenbeger, 1976, Patton, 2004, Narayan and Ahmed, 2014) shows that the presence of moments higher than the second order influences investors mean and variance utility function. It follows that investor preferences for higher moments are important for asset pricing. For example, investors have positive preferences for third and higher odd moments and negative preferences for second and higher even moments. Given these preferences, higher order moments are likely to influence asset pricing which if they do should be reflected in price forecasts. Specifically, our strategy in this paper is motivated by a generalised autoregressive conditional heteroscedasticity (GARCH) model, which we augment with skewness (S) and kurtosis (K). We refer to this as a GARCH-SK model. We apply this model to a popular (and extensive) dataset on the United States. This dataset, widely used in empirical applications in this literature, contains United States stock returns as well as 19 popular predictors of returns. Commonly referred to as the Welch and Goyal (WG, 2008) dataset, it offers an ideal resource to compare predictive regression models.

Our findings can be summarised as follows. First, our proposed GARCH-SK model, in terms of in-sample predictability, beats a model without SK (namely the WG predictive regression model). For example, we have at most 19 predictors and so a maximum of 19 predictive regression models. With the GARCH-SK model, we find stronger evidence of predictability compared to the WG model. Using annual, quarterly and monthly data, we find that with the GARCH-SK model 42%, 72% and 75% of predictors, respectively, predict returns. By comparison, with the WG model, predictability is found in 26% (annual), 55% (quarterly) and 69% (monthly) of cases. We, therefore, find that the GARCH-SK model beats the WG model at all data frequencies, with the performance stronger at the annual and quarterly data frequencies.

Second, the out-of-sample results are in general more in favour of the GARCH-SK model. For annual data, although most variables are unable to predict returns, we find GARCH-SK model performs better than the WG model with respect to three predictors. The results at the quarterly frequency are strongly in favour of GARCH-SK model: it outperformances the WG model with respect to 11 predictors. Lastly, when using monthly data, we find six variables are statistically stronger predictors only when using GARCH-SK model.

Third, when we extend the out-of-sample test to an economic significance test based on computing investor utilities for a mean-variance investor, the evidence in favour of the GARCH-SK model is remarkable. With annual data (BM, DFR, INFL, and EQIS), quarterly data (BM, TBL, DFY, DFR, KTY, NTIS, CORPR, and CSP), and monthly data (DP, DY, EP, BM, TBL, DFY and INFL) investor utilities are greater when forecasts are generated using the GARCH-SK model compared to the WG model. On the whole while both statistically and economically, the GARCH-SK model better, the economic significance results are much more favourable to the GARCH-SK model.

Our results contribute to the menu of econometric methods available to researchers for undertaking predictability tests. In this regard, our paper complements studies such as Westerlund and Narayan (2015), who show the importance of heteroscedasticity in predictability; Paye and Timmermann, 2006, Lettau and Van Nieuwerburgh, 2008, who show, respectively, the role of structural breaks in the coefficients of the predictive regression and the presence of shifts in the predictor variable; and Henkel et al., 2011, Devpura et al., 2018, who focus on the concept of time-variation in predictability. The main message emanating from these studies is that predictability is indeed influenced by the approach one takes. The story emerging from our contribution is similar. An added advantage of our proposed method is that it can be used to test for predictability of many other financial variables, namely exchange rates, interest rates, bond returns, commodity market returns, credit default swap spreads, amongst others. The key message of our work is that as long as one is interested in forecasting data which has skewness and kurtosis features (that need modelling, and many high frequency data fit this category), our proposed model will be useful.

We organise the rest of the paper as follows. In the next section we outline the methodology. Section 3 contains results while the final section provides concluding remarks.

Section snippets

Model description

The starting point for our test of stock return predictability is a predictive regression framework of the following form:rt=a0+a1xt-1+εtwhere rt denotes stock return or the equity premium, xt-1 is the independent variable used to predict the equity premium. We consider an asset return model with the GARCH(1,1) structure for conditional variance and also a GARCH(1,1) structure for both conditional skewness and kurtosis. We have a total of 19 predictors as described in WG and as listed in Table 1

Data and results

We have a range of results from tests of our GARCH-SK model vis-à-vis the WG model. The first set relates to in-sample predictability. The in-sample test is based on testing the null hypothesis that there is predictability. We have 19 predictors (as noted in column 1 of Table 1), therefore, we have 19 predictive regression models. These regressions are run using our GARCH-SK model and the WG non-GARCH model. The second set is about the out-of-sample performance. We use two metrics here, namely,

Concluding remarks

This paper is motivated by a growing focus on testing for stock return predictability. There has been a growing interest in methodological issues surrounding testing for predictability and the application of such models. Our contribution is both methodological and empirical in the sense that we propose a model that accommodates higher order moments (namely, skewness and kurtosis) in testing the null hypothesis of no predictability. Using an extensive time-series dataset on financial and

References (31)

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