Modelling returns in US housing prices: You're the one for me, fat tails

Kiss, Tamás; Nguyen, Hoang; Österholm, Pär Article Modelling returns in US housing prices: You're the one for me, fat tails Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Kiss, Tamás; Nguyen, Hoang; Österholm, Pär (2021) : Modelling returns in US housing prices: You're the one for me, fat tails, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 14, Iss. 11, pp. 1-17, https://doi.org/10.3390/jrfm14110506 This Version is available at: https://hdl.handle.net/10419/258610 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Modelling Returns in US Housing Prices—You’re the One for Me, Fat Tails Tamás Kiss 1, Hoang Nguyen 2and Pär Österholm 1,3,*   Citation: Kiss, Tamás, Hoang Nguyen, and Pär Österholm. 2021. Modelling Returns in US Housing Prices—You’re the One for Me, Fat Tails. Journal of Risk and Financial Management 14: 506. https:// doi.org/10.3390/jrfm14110506 Academic Editor: Goran Buturac Received: 17 September 2021 Accepted: 15 October 2021 Published: 20 October 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1Division of Economics, School of Business, Örebro University, 701 82 Örebro, Sweden; [email protected] 2Division of Statistics, School of Business, Örebro University, 701 82 Örebro, Sweden; [email protected] 3National Institute of Economic Research, 102 23 Stockholm, Sweden *Correspondence: par.oster[email protected] Abstract: In this paper, we analysed the heavy-tailed behaviour in the dynamics of housing-price returns in the United States. We investigated the sources of heavy tails by estimating autoregressive models in which innovations can be subject to GARCH effects and/or non-Gaussianity. Using monthly data from January 1954 to September 2019, the properties of the models were assessed both withinand out-of-sample. We found strong evidence in favour of modelling both GARCH effects and non-Gaussianity. Accounting for these properties improves within-sample performance as well as point and density forecasts. Keywords: non-Gaussianity; GARCH; probability integral transform; Kullback–Leibler information criterion JEL Classification: C22; C52; E44; E47; G17 1. Introduction In less than fifteen years, the world has experienced both a global financial crisis and a virus pandemic—both of which have had dire economic consequences. These events have confirmed the fact that large swings in economic variables happen more frequently than what is implied by models based on the traditional assumption of normally distributed disturbances. Put differently, the distributions of many variables are characterised by fat tails (Fagiolo et al. 2008;Ascari et al. 2015), that is, they have a higher probability mass in the tails of the distribution than a normal distribution does. The purpose of this paper is to study the topic of fat tails related to a key US variable, namely, housing-price returns, an important macro-finance variable. Our analysis focuses on the dynamic properties of the process. We primarily aimed to assess what the appropriate distributional features of the innovations are if one aims to model the dynamics of the returns. We studied this issue using different autoregressive (AR) models, where we allowed for different sources of non-Gaussianity. We first conducted within-sample analysis and then validated our findings through an out-of-sample forecast exercise. Our analysis was performed by employing monthly data on US housing-price returns from January 1954 to September 2019. We confirmed that the return series was characterised by fat tails and accordingly investigated its sources. We assessed the relevance of timevarying volatility—which can be a source of fat tails in the unconditional distribution of the variable—by estimating GARCH models. Whether error terms are drawn from distributions with fat tails was addressed by abandoning the assumption of a normal distribution for that of a Laplace distribution or a Student-tdistribution. Finally, since the unconditional distribution of the data also appeared to be somewhat skewed, we also estimated the model under the assumption that error terms are drawn from a Skew-t distribution (defined as in Hansen 1994). After summarising our results, we found strong in-sample evidence for highly persistent, time-varying volatility and heavier-than-Gaussian J. Risk Financial Manag. 2021,14, 506. https://doi.org/10.3390/jrfm14110506 https://www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2021,14, 506 2 of 17 tails. This translates into improved short-term point and density forecasts, with GARCH effects being the most important feature. Skewness, on the other hand, appears to be a less relevant aspect to consider in modelling. We contribute to the literature in two distinct ways. First, we provide further evidence on non-Gaussianity of housing returns; see, for example, Myer and Webb (1994); Young and Graff (1995); Maurer et al. (2004) and Pontines (2010). In our analysis we focus on the dynamic and forecasting properties of the housing returns and our results suggest that deviations from the normal distribution do matter for forecasting purposes, that is, for the distribution of the innovations hitting housing returns. Second, we add to the literature analysing non-Gaussian behaviour in macroeconomic variables (e.g., Fagiolo et al. 2008 ; Ascari et al. 2015) by adding evidence on housing returns. Related to this issue, our findings echo the need to incorporate non-Gaussian behaviour when modelling and forecasting macroeconomic variables. The rest of this paper is organised as follows: Section 2includes a brief review of the related academic literature. In Section 3, we present the data. The methodological framework is discussed in Section 4and our empirical findings are shown in Section 5. Finally, Section 6concludes the paper. 2. Related Literature Housing prices are not only important as asset prices, but they are also highly relevant from a macroeconomic perspective since they have a close relation to the real economy. Several studies have documented their connection with the business cycle in general. Iacoviello (2005) used a general equilibrium monetary policy model with nominal loans and housing used as collateral. His main conclusion was that the collateral effect improves aggregate demand response to housing prices. In a similar spirit, Iacoviello and Neri (2010) showed—also using a general equilibrium framework—that housing-market spillovers are non-negligible and that these effects are concentrated on consumption rather than aggregate investment. Klarl (2016) employed wavelet (frequency) analysis to find that housing prices only lead the economic cycle during recessions, and shocks associated with the housing market are relatively short-lived. The relationship between housing wealth and consumption has received particular attention. Catte et al. (2004) focused on factors of housing-price variability. By estimating error correction models for OECD countries, they found that both macroeconomic and structural factors contribute to housing-price variability and that the propensity to consume depends on how developed the mortgage market is in a given country. Campbell and Cocco (2007) analysed household data from the United Kingdom and found heterogeneity in the propensity of consumption to housing wealth. In particular, older households are more sensitive to housing-price changes and regional housing-price variation is more important than national variation. A similar conclusion was reached by Martín-Legendre et al. (2019) using Spanish household survey data, who also looked at other asset classes and found that the propensity to consume is highest for housing equity. Acolin (2020) found time variation in the relationship between housing prices and consumption based on survey data from the United States. In particular, the propensity to consume increases right before the financial crisis and decreases afterwards. He argues that these changes are associated with the availability of tools to extract wealth from home equity. The importance of wealth extraction is also pointed out by Pan and Wu (2021), who analysed a panel of Chinese homeowners and found that the propensity to consume from housing wealth concentrates on second houses, and not on primary residences. There are a number of studies that document non-Gaussian behaviour in real estateprice changes. These papers mostly document the presence of skewness and kurtosis in the unconditional distribution of both nominal and real (adjusted for inflation) housingprice returns, which is important from a portfolio risk-management perspective where housing equity is considered as a financial asset; see, for example, Foster and Van Order (1984) who used stable distribution in prices to model volatility and default risk. Myer J. Risk Financial Manag. 2021,14, 506 3 of 17 and Webb (1994) considered both real and nominal returns at a monthly, quarterly and annual frequency. They found that non-Gaussianity is stronger in nominal returns and at a quarterly (rather than monthly or annual) frequency. Young and Graff (1995) applied stable distributions for commercial real estate indices in the United States and found that models with infinite variance (very heavy tails) are better suited than models based on the Gaussian distribution. A similar analysis was performed by Graff et al. (1997) for Australian data and Young et al. (2006) for data from the United Kingdom, with the conclusion that infinite variance models fit the data in the respective countries better. For Germany, Maurer et al. (2004) analysed returns on real estate fund indices and found relatively less kurtosis and skewness than in the United States. Leung et al. (2006) analysed the macroeconomic factors affecting the second moment and skewness of housing returns and found relatively weak support for macroeconomic factors to drive housing-price dispersion and skewness. Pontines (2010) found evidence for fat tails in 16 OECD countries and showed that the Student t-distribution is adequate to fit almost all data. Chiang et al. (2012) argue that value-at-risk based on the Gaussian distribution is inappropriate since the downside risk of housing prices is larger than implied by the Gaussian distribution; it is actually described well by a so-called generalised Pareto distribution for both real and nominal returns in the United States and the United Kingdom. Our analysis with respect to non-Gaussianity focuses on the dynamic properties of the housing-price process; in that sense, it is related to Dufitinema (2021), who assessed Bayesian stochastic volatility models and found that the model with heavy tails performs the best in terms of out-of-sample forecasts for several Finnish housing-price series. Dynamic interactions were also the focus of McMillan (2012), who considered the long-run relationship between stock prices and housing prices in the Unites States and the United Kingdom with an error correction model. He found that the adjustment to equilibrium is non-linear; large deviations must occur before stock prices revert, while housing prices hardly adjust at all. Musso et al. (2011) estimated a structural VAR for housing prices and business-cycle variables and argue that housing-price shocks are not identified using the standard Cholesky decomposition in a model with Gaussian disturbances. Ahamada and Sanchez (2013) assessed the role of housing-price dynamics in a small-scale macroeconomic VAR model and found evidence that the effect of housing-prices on economic activity intensified during the 1980s. Housing-price dynamics are also affected by market efficiency and potential nonrational behaviour. Case and Shiller (1989) were pioneers in documenting that when housing markets are inefficient, there is a positive autocorrelation in housing returns and real interest rate information does not seem to be fully incorporated in house prices. Granziera and Kozicki (2015) solved an asset pricing model for housing prices with bubbles created by waves of over-optimism, and they calibrated the model to data in the United States. Their results show that this model captures housing-price dynamics better than the rational expectations model. Analysing household survey data, Ma (2020) found support for non-rational behavior. In particular, households’ housing-price expectations tend to exhibit momentum (positive serial correlation), and no significant reversion to fundamentals. Lastly, our paper contributes to the literature on non-Gaussianity in macroeconomic variables. This issue has received increasing attention in the aftermath of the financial crisis and the recent economic downturn associated with the coronavirus pandemic. Fitting exponential-power distributions, Fagiolo et al. (2008) found that the output-growth in several OECD countries has heavier than normal tails, even after controlling for autocorrelation, heteroscedasticity and outliers. Related to that, Ascari et al. (2015) simulated standard new-Keynesian and real business cycle models, and they suggest that these models may prove to be unable to generate the heavy tails in the unconditional distribution documented in output growth. Skewness—that is, asymmetry—has also been discussed for macroeconomic variables. Neftci (1984) has documented that the unemployment rate is asymmetric by estimating a finite state Markov process. Time series evidence based on J. Risk Financial Manag. 2021,14, 506 4 of 17 aggregate data in Acemoglu and Scott (1997) also suggests that output in the United States is best captured by non-linear, asymmetric models where sharper downturns (a heavier left tail) are allowed for. Lastly, Bekaert and Popov (2019) used panel data from 110 countries to document a positive cross-sectional correlation between the volatility and skewness of output growth. 3. Data We based our analysis on Shiller’s (2015) housing-price index. In our main analysis, we used nominal housing prices, in line with several previous studies, including Young and Graff (1995); Young et al. (2006); Pontines (2010) and Ma (2020). Nominal returns are also the focus of some important surveys related to housing prices, such as the University of Michigan’s Survey of Consumers; for analysis based on this survey, see, for example, Piazzesi and Schneider (2009) and Hjalmarsson and Österholm (2020). However, since real housing returns (adjusted for inflation) are also of interest and have been analysed in the literature, we also consider real returns in Appendix B; 1 both nominal and real returns are considered in, for example, Case and Shiller (1989); Myer and Webb (1994); Maurer et al. (2004); Chiang et al. (2012) and Jordáet al. (2019). Data are monthly and span the period from January 1953 to September 2019. Nominal returns are calculated as rt=Pt−Pt−12 Pt−12 , where Pt is the price index at time t. A time-series plot of the return data is given in Figure 1and a histogram showing the unconditional distribution is provided in Figure 2. Some key descriptive statistics are given in Table 1. J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 4 of 18 discussed for macroeconomic variables. Neftci (1984) has documented that the unemployment rate is asymmetric by estimating a finite state Markov process. Time series evidence based on aggregate data in Acemoglu and Scott (1997) also suggests that output in the United States is best captured by non-linear, asymmetric models where sharper downturns (a heavier left tail) are allowed for. Lastly, Bekaert and Popov (2019) used panel data from 110 countries to document a positive cross-sectional correlation between the volatility and skewness of output growth. 3. Data We based our analysis on Shiller’s (2015) housing-price index. In our main analysis, we used nominal housing prices, in line with several previous studies, including Young and Graff (1995); Young et al. (2006); Pontines (2010) and Ma (2020). Nominal returns are also the focus of some important surveys related to housing prices, such as the University of Michigan’s Survey of Consumers; for analysis based on this survey, see, for example, Piazzesi and Schneider (2009) and Hjalmarsson and Österholm (2020). However, since real housing returns (adjusted for inflation) are also of interest and have been analysed in the literature, we also consider real returns in Appendix B;1 both nominal and real returns are considered in, for example, Case and Shiller (1989); Myer and Webb (1994); Maurer et al. (2004); Chiang et al. (2012) and Jordá et al. (2019). Data are monthly and span the period from January 1953 to September 2019. Nominal returns are calculated as =  , where  is the price index at time t. A time-series plot of the return data is given in Figure 1 and a histogram showing the unconditional distribution is provided in Figure 2. Some key descriptive statistics are given in Table 1. Figure 1. Housing–price return data. Note: Twelve–month change in percent on vertical axis. Figure 1. Housing–price return data. Note: Twelve–month change in percent on vertical axis. Table 1. Descriptive statistics of the returns, Jarque–Bera and unit root test statistic. Mean Variance Skewness Kurtosis Jarque–Bera ADF KPSS N 4.278 24.390 −0.150 3.813 24.464 −3.505 0.166 789 Note: N is the number of observations. The critical value at the five percent level of the Jarque–Bera test is 5.99. The Augmented Dickey–Fuller (ADF) test is performed with four lagged differences in the test equation (to be consistent with the selected lag length for the empirical model considered in Section 5). Both unit-root tests are performed under the assumption that there is no deterministic trend in the data (the test equation has an intercept but no time trend). The critical value at the five percent level for the ADF and KPSS test is − 2.866 and 0.463, respectively. J. Risk Financial Manag. 2021,14, 506 5 of 17 J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 5 of 18 Figure 2. Histogram of housing–price return data. Note: Twelve–month change in percent on horizontal. Relative frequency on vertical axis. Table 1. Descriptive statistics of the returns, Jarque–Bera and unit root test statistic. Mean Variance Skewness Kurtosis Jarque–Bera ADF KPSS N 4.278 24.390 −0.150 3.813 24.464 −3.505 0.166 789 Note: N is the number of observations. The critical value at the five percent level of the Jarque–Bera test is 5.99. The Augmented Dickey–Fuller (ADF) test is performed with four lagged differences in the test equation (to be consistent with the selected lag length for the empirical model considered in Section 5). Both unit-root tests are performed under the assumption that there is no deterministic trend in the data (the test equation has an intercept but no time trend). The critical value at the five percent level for the ADF and KPSS test is −2.866 and 0.463, respectively. As both the histogram in Figure 2 and the descriptive statistics in Table 1 suggest, the tails of the distribution appear heavier than what is implied by a normal distribution; the kurtosis of the unconditional distribution is around 3.8. There is also a slight negative skewness. Taken together, this means that the Jarque–Bera test strongly rejects an assumption of normality. Furthermore, both the Augmented Dickey–Fuller test and the KPSS test strongly suggest that the return series does not contain a unit root. In Table 2, we compare the unconditional housing-price returns and a normally distributed variable using the kurtosis (,), peakedness ( ,) and tailness ( ,) measures proposed by Liu (2019). These measures are based on the ratio of interquantile ranges of the variables, using various quantiles of the distribution denoted by , , and  (for example,  = 0.05 means that we use the 5th percentile of the distribution when calculating kurtosis and tailness).2 Of particular importance are the tails of the return distribution, which are heavier than those of the normal distribution at all levels. There is also some evidence for asymmetry in the unconditional distribution further out in the tails (α = 0.01), but skewness is not a robust feature if we use different quantiles (α). Table 2. Comparison of quantile–based kurtosis, peakedness and tailness.  , , , , , Normal distribution 0.010 3.449 1.706 2.022 1.011 1.011 0.025 2.906 1.706 1.704 0.852 0.852 Figure 2. Histogram of housing–price return data. Note: Twelve–month change in percent on horizontal. Relative frequency on vertical axis. As both the histogram in Figure 2and the descriptive statistics in Table 1suggest, the tails of the distribution appear heavier than what is implied by a normal distribution; the kurtosis of the unconditional distribution is around 3.8. There is also a slight negative skewness. Taken together, this means that the Jarque–Bera test strongly rejects an assumption of normality. Furthermore, both the Augmented Dickey–Fuller test and the KPSS test strongly suggest that the return series does not contain a unit root. In Table 2, we compare the unconditional housing-price returns and a normally distributed variable using the kurtosis ( Kα,τ ), peakedness ( Pη,τ ) and tailness ( Tα,η ) measures proposed by Liu (2019). These measures are based on the ratio of interquantile ranges of the variables, using various quantiles of the distribution denoted by α , η , and τ (for example, α= 0.05 means that we use the 5th percentile of the distribution when calculating kurtosis and tailness). 2 Of particular importance are the tails of the return distribution, which are heavier than those of the normal distribution at all levels. There is also some evidence for asymmetry in the unconditional distribution further out in the tails ( α = 0.01), but skewness is not a robust feature if we use different quantiles (α). Table 2. Comparison of quantile–based kurtosis, peakedness and tailness. αKα,τPη,τTη,τRTη,τLTη,τ Normal distribution 0.010 3.449 1.706 2.022 1.011 1.011 0.025 2.906 1.706 1.704 0.852 0.852 0.050 2.439 1.706 1.430 0.715 0.715 Unconditional returns 0.010 3.979 1.404 2.834 1.256 1.579 0.025 3.317 1.404 2.363 1.141 1.222 0.050 2.615 1.404 1.862 1.013 0.850 Note: Comparison of the quantile–based kurtosis, peakedness and tailness among a normal random variable and the unconditional housing–price returns. The measurements are calculated based on ratios of interquantile ranges with α= (0.01, 0.025, 0.05) and η= 0.125, τ= 0.25 quantile levels (see Footnote 2 and Liu 2019). 4. Methodological Framework Since the focus of our analysis was to model the properties of the housing-return series, we used univariate time series to capture the dynamics of the returns. We used AR models because they typically perform well in macroeconomic forecasting (e.g., Faust and Wright 2009) , J. Risk Financial Manag. 2021,14, 506 6 of 17 despite their simplicity and parsimony. Our baseline model is the homoscedastic univariate AR(p) model: rt=γ0+ p ∑ j=1 γjrt−j+σννt(1) where νt is assumed to be an iid normal error term, νt∼N(0, 1) . The number of lags is determined using the Schwarz (1978) information criterion, which suggests a lag length of p= 5 ; for comparability we employed the same number of lags across all model specifications. We then allowed for a more flexible error term in the model. First, we let the second moment vary over time. Doing that, we choose a robust approach and rely on the GARCH(1,1) specification (Bollerslev 1986), with a normally distributed error term. This is a common modelling choice—see Chung et al. (2012) and Clark and Ravazzolo (2015) among others for macroeconomic variables—and for financial data, a GARCH(1,1) model is fairly difficult to outperform (Hansen and Lunde 2005). Hence, the second model was the AR(p)-GARCH(1,1): rt=γ0+ p ∑ j=1 γjrt−j+et(2) et=σtvt(3) σ2 t=ω+α1e2 t−1+α2σ2 t−1(4) where vt , as above, is assumed to be an iid, normal error term. In the third and fourth models, we relax the assumption of normality of the error term, while maintaining symmetry. This is done by using νt∼Laplace and νt∼Student −t(κ) , where κ is the degrees of freedom. Of specific interest here is the fact that both the Laplace distribution and the Student-tdistribution allow for heavier-than-Gaussian tails. 3 Finally, in the fifth model, we assume that the error terms are drawn from the Skew-tdistribution (Hansen 1994)—that is, νt∼Skew −t(λ,κ) , where κ is the degrees of freedom and λ is the skewness parameter. This means that we allowed for both heavy tails and asymmetry in the error term.4 Our primary focus was to evaluate which distributional properties are important for modelling the dynamics of the housing-price return series. We evaluated this based on both within-sample estimates (using maximum likelihood) and an out-of-sample forecasting exercise; in the latter, we considered both point and density forecasts. The point forecasts were evaluated based on the root mean square error (RMSE) of the forecast, accompanied by the Diebold–Mariano test (Diebold and Mariano 1995). We formulated the test such that a positive test value implies that the given model outperforms the benchmark.5 In terms of density forecast evaluation, we made use of the probability integral transform (PIT) proposed by Diebold et al. (1998). We looked at the histograms of the PIT and visually assessed how close they are to the uniform distribution. We also employed more rigorous statistical techniques such as the Kullbach–Leibler (KL) divergence (Kullback and Leibler 1951) of the PIT series from an iid uniform distribution and formal statistical tests of uniformity, namely, the Kolmogorov–Smirnov (KS) and the Anderson–Darling (AD) test (Anderson and Darling 1952). 6 For all these measures, a lower value implies that the given model produces a better density forecast. 5. Empirical Analysis We start by presenting within-sample estimation results for all five specifications. 7,8 After that, we turn to an out-of-sample forecast exercise. In the forecast exercise, we focus on the one-period-ahead horizon. The main reason for this is that results for that horizon theoretically are most closely related to within-sample findings and can hence be considered a validation tool. J. Risk Financial Manag. 2021,14, 506 7 of 17 5.1. Within-Sample Estimation Results Table 3presents the estimation results for the five specifications. Note that the housingprice return series is fairly persistent (the sum of the autoregressive coefficients are close to unity). Engle’s (1982) ARCH test shows the strong presence of conditional heteroscedasticity for the residuals in the baseline model in column 1. Also, the Jarque–Bera test suggests that residuals are non-Gaussian in this specification. This is further supported by Figure 3, where it is clear that a t-distribution fits the residuals of the baseline specification better than a normal distribution. Table 3. Within–sample estimation results. (1) (2) (3) (4) (5) Mean equation [AR(5)] γ00.038 0.033 0.035 0.033 0.032 (0.023) (0.185) (0.000) (0.467) (0.013) γ11.564 1.816 1.823 1.820 1.819 (0.000) (0.000) (0.000) (0.000) (0.000) γ2−0.509 −0.756 −0.743 −0.741 −0.740 (0.000) (0.000) (0.000) (0.023) (0.000) γ3−0.115 −0.271 −0.321 −0.307 −0.303 (0.103) (0.011) (0.000) (0.147) (0.020) γ40.203 0.396 0.415 0.400 0.397 (0.003) (0.209) (0.000) (0.000) (0.006) γ5−0.152 −0.189 −0.180 −0.178 −0.178 (0.000) (0.263) (0.000) (0.001) (0.003) GARCH equation ω0.000 0.000 0.001 0.001 (0.745) (0.260) (0.895) (0.702) α10.172 0.123 0.175 0.175 (0.459) (0.001) (0.824) (0.671) α20.828 0.804 0.825 0.825 (0.000) (0.000) (0.000) (0.001) Degrees of freedom κ7.870 7.941 [26.854] [3.297] Skewness parameter λ−0.031 (0.854) ARCH-test 131.592 13.661 14.431 13.440 13.494 (0.000) (0.322) (0.274) (0.338) (0.334) JB-test 295.989 51.528 15.288 a0.616 a0.070 a (0.000) (0.000) (0.000) (0.735) (0.966) N 789 789 789 789 789 Note: Model (1) is a homoscedastic model with Gaussian error terms. Model (2) employs a GARCH(1,1) specification with Gaussian error terms. Models (3), (4) and (5) employ a GARCH(1,1) specification together with Laplace, t–, and Skew–t–distributed error terms, respectively. The ARCH-test is Engle’s test for conditional heteroscedasticity (Engle 1982) conducted with twelve lags. The JB–test is the Jarque–Bera test for conditional heteroscedasticity. “ a ” indicates that the test is based on the probability integral transform of the (standardised) residuals (see footnote 10). p–values are in parentheses (); standard errors for the degrees of freedom of the t-distribution are in brackets []. “N” is the number of observations. The GARCH(1,1) specification (column 2) seems to take care of conditional heteroscedasticity quite well. However, the variance is very persistent and its parameters are not precisely estimated. 9 For the specification with Laplace distribution, where no extra parameter is added to capture heavy-tails, the persistence of the variance is somewhat lower. However, it seems that the Laplace distribution is not flexible enough to capture the behaviour of the standardised residuals, as their transformation by imposing a normal distribution still shows signs of non-Gaussianity according to the Jarque–Bera test. 10 In contrast, the more flexible approaches, the tand the Skew-tdistribution, seem appropriate to capture non-Gaussianity. However, the skewness parameter is low and non-significant (column 5). J. Risk Financial Manag. 2021,14, 506 8 of 17 J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 8 of 18 N 789 789 789 789 789 Note: Model (1) is a homoscedastic model with Gaussian error terms. Model (2) employs a GARCH(1,1) specification with Gaussian error terms. Models (3), (4) and (5) employ a GARCH(1,1) specification together with Laplace, t–, and Skew–t–distributed error terms, respectively. The ARCH-test is Engle’s test for conditional heteroscedasticity (Engle 1982) conducted with twelve lags. The JB–test is the Jarque–Bera test for conditional heteroscedasticity. “a” indicates that the test is based on the probability integral transform of the (standardised) residuals (see footnote 10). p– values are in parentheses (); standard errors for the degrees of freedom of the t-distribution are in brackets []. “N” is the number of observations. Figure 3. Residuals from homoscedastic model. Note: Density on vertical axis. The GARCH(1,1) specification (column 2) seems to take care of conditional heteroscedasticity quite well. However, the variance is very persistent and its parameters are not precisely estimated.9 For the specification with Laplace distribution, where no extra parameter is added to capture heavy-tails, the persistence of the variance is somewhat lower. However, it seems that the Laplace distribution is not flexible enough to capture the behaviour of the standardised residuals, as their transformation by imposing a normal distribution still shows signs of non-Gaussianity according to the Jarque–Bera test.10 In contrast, the more flexible approaches, the t and the Skew-t distribution, seem appropriate to capture non-Gaussianity. However, the skewness parameter is low and non-significant (column 5). Figure 4 shows the estimated conditional volatility of the return series. All GARCH specifications look quite similar; the biggest discrepancy is found for the Laplace, which due to its excessively heavy tails allows for a generally lower variance. Volatility also clusters; the series starts out with a higher general level of volatility, is considerably lower between 1980 and 2005, and peaks again during the financial crisis. Having assessed the different specifications within-sample, we next conducted an out-of-sample forecast exercise to validate our results. If we can establish with this exercise that a more complex specification also generates better out-of-sample forecasts, we can conclude that our results are not due to over-fitting the data within-sample. Figure 3. Residuals from homoscedastic model. Note: Density on vertical axis. Figure 4shows the estimated conditional volatility of the return series. All GARCH specifications look quite similar; the biggest discrepancy is found for the Laplace, which due to its excessively heavy tails allows for a generally lower variance. Volatility also clusters; the series starts out with a higher general level of volatility, is considerably lower between 1980 and 2005, and peaks again during the financial crisis. J. Risk Financial Manag. 2021, 14, x FOR PEER REVIEW 9 of 18 Figure 4. Conditional volatility under different assumptions for the error term. Note: Conditional standard deviation on vertical axis. 5.2. Out-of-Sample Analysis We evaluated the one-period-ahead forecasting performance of the models using an expanding window technique starting in January 1974; this yielded F = 548 forecasts to evaluate.11 The results are collected in Table 4 for both point and density forecast measures. Table 4. Forecast evaluation results. RMSE DM KS AD KL Normal—Homoscedastic 0.279 - 0.151 25.498 0.179 Normal—GARCH 0.259 3.491 0.044 1.743 0.013 Laplace—GARCH 0.258 3.394 0.101 13.141 0.066 Student-t—GARCH 0.258 3.495 0.036 0.590 0.008 Skew-t—GARCH 0.258 3.468 0.035 0.521 0.006 Note: “RMSE” is the root mean squared error of the point forecasts. “DM” is the Diebold–Mariano test statistic where the model with homoscedastic, normally distributed errors is chosen to be the benchmark. The critical value at the five percent level for the Diebold–Mariano test is 1.65. “KS” is the Kolmogorov–Smirnov test statistic. “AD” is the Anderson–Darling test statistic. “KL” is the Kullback–Leibler divergence of the PIT of the residuals from the uniform distribution on the [0,1] interval. All three density forecast measures are based on the PIT of the density forecasts produced by each model. The results are based on F = 548 out-of-sample forecasts and an expanding window for parameter estimation. The critical value at the five percent level for the Kolmogorov–Smirnov test is 0.058 and for the Anderson–Darling test it is 2.492. For point forecasts, the homoscedastic model clearly performs worst. Accounting for time-varying volatility statistically significantly improves the forecast, as can be seen from the Diebold–Mariano test. However, allowing for non-Gaussian error terms does not provide additional improvement in point forecasts over the Gaussian GARCH specification. For density forecasts, the homoscedastic benchmark model again appears to have the weakest performance. Not only are all three evaluation measures largest for this model, the KS and AD tests reject uniformity of the PIT series at the five percent level. Comparing Figure 4. Conditional volatility under different assumptions for the error term. Note: Conditional standard deviation on vertical axis. Having assessed the different specifications within-sample, we next conducted an out-of-sample forecast exercise to validate our results. If we can establish with this exercise that a more complex specification also generates better out-of-sample forecasts, we can conclude that our results are not due to over-fitting the data within-sample. J. Risk Financial Manag. 2021,14, 506 15 of 17 2The measures are related to each other and the quantiles of the distribution by the formula Kα,τ=Pη,τTα,η=Q1−η−Qη Q1−τ−Qτ Q1−α−Qα Q1−η−Qη , where Qi is the i -th quantile of the distribution and the quantiles used satisfy 0 <α<η<τ< 0.5. Further, the tail ( T ) can be decomposed to right (RT) and left (LT) tails according to Tα,η=RTα,η+LTα,η=Q1−α−Q0.5 Q1−η−Qη +Q0.5 −Qα Q1−η−Qη. 3 Fagiolo et al. (2008) found the Laplace distribution useful when modelling the fat tails of GDP growth rates. The Student-t distribution has been used more widely in the empirical literature; see, for example, Cúrdia et al. (2014); Clark and Ravazzolo (2015) ; Cross and Poon (2016); and Kiss and Österholm (2020). 4 All distributions are parameterised to have a unit variance. The density function of the Skew-t distribution can be found in Hansen (1994). 5 As pointed out by Diebold (2015), the Diebold and Mariano (1995) test in its standard form is a reasonable choice even if we employ nested models, for which the original assumptions of the test do not formally hold. This is further supported by the fact that the test performs relatively well in such larger training and evaluation samples as the one we use in our analysis (Clark and McCracken 2013). 6 The KL divergence has been used in a number of applications in the context of density forecasts—typically though with a somewhat different focus; see, for example, Cogley et al. (2005); Robertson et al. (2005); Hall and Mitchell (2007); Diks et al. (2010) and Mitchell and Wallis (2011). 7 Figure 1suggests that the time-series behaviour of the return series may have changed around 1975. In fact, the Shiller (2015) dataset changes source in January 1975. Therefore, we also estimate the model on the shorter subsample, namely, January 1975 to September 2019. The results collected in Table A1 in Appendix Aare qualitatively very similar to full sample estimates. 8 We also assessed the robustness of our results to the presence of different regimes, in particular the boom and bust cycle between 2002 and 2009. We did this by allowing for a different constant and dynamics in the mean equation during this period, using a time dummy and interactions. Unreported results (available upon request from the authors) show very similar results to our baseline. Capturing non-Gaussianity in the innovations remains a salient and important feature of the model. 9 In fact, the parameters α1 and α2 sum up to unity for the normal-GARCH, Student-t-GARCH and Skew-t-GARCH specifications. However, looking at the results using the Laplace distribution and the shorter sample, integrated volatility does not seem to be a robust feature of the data, therefore we do not impose it in any of the specifications with conditional heteroscedasticity. 10 The Jarque–Bera tests are based on zt , the PIT series of the standardised residuals, for which Φ−1(zt) is standard normally distributed (where Φ−1 is the inverse cumulative distribution function of the standard normal distribution). We test this by applying the Jarque–Bera test on Φ−1(zt). 11 That is, we first estimate the models on the sample January 1953 to January 1974 and make predictions for February 1974. 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