Univariate analysis

The sample autocorrelation function (ACF) and the partial autocorrelation function (PACF) for each asset-return provide a rough indication, for our univariate time series, of whether the DGP has stationarity or not. The corrologram figures, not shown here, indicate that it is not likely for any of our asset-return samples to be generated by a white-noise process, because some of the sample autocorrelations and sample partial autocorrelations cross the dotted lines. However, lots of the coefficients at higher lags are clearly between the dashed lines. Hence, the underlying ACF of most of the countries’ asset-return may be in line with a stationary DGP.

Although the ACF and PACF provide an initial test of stationarity, their indication is not evident. Therefore, we applied proper statistical tests for stationarity. One form of testing a stationarity assumption is to test for a unit root. Each of the widely applicable tests of unit roots (a form of non-stationarity) has been applied on each of our asset-return independently, namely the Augmented Dickey-Fuller (ADF) test, the Phillips-Perron (PP) test, and the

3.7 The initial step: Data analysis of asset-returns and their predicted DGP 111

Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test. In our series, we observe some degree of disagreement among the results of these tests. For example, with a 5% significance level, we found that France Real Equity return series suffers from a unit root using the ADF test, but this is not the result using the PP test and the KPSS test. However, in general, we may conclude that all our series are weakly stationary6. However, we will rely more on the ADF test, because it is more robust than other tests when applied to data-sets that might be correlated or have witnessed some heavy-tail phenomena (see,Wang & Mao 2008).

In the wake of our previous analysis, it seems that our sample of asset returns for all our sample countries is assumed to be stationary. Recall that the data used are described in Section3.6. These data-sets are annual data; hence, also known as low-frequency data as opposed to the high-frequency data-set. It is valid to argue that since each data point in our data-set is recorded based on annual observation, the dependence structure changes as opposed to data points that are recorded based on a day-by-day basis. Therefore, it would be natural to reject, based on our sample data, that the asset returns originate from a GARCH type time series process.

To conclude, these data-sets may be originating from an Auto-Regressive Moving Average (ARMA) models. Moreover, we are keen to test whether the data at hand suffers from time- variation in the variance. However, before testing for any Auto-Regressive Conditionally Heteroskedastic (ARCH) effect in our asset return series, we would need to ensure that we have taken advantage of all the information we hold to analyse our univariate series. Particularly, we would like to check whether a country’s equity-returns play any significant role in predicting the same country’s bond returns, or vice-verse.

The sample cross-correlation function (CCF) is a helpful tool for identifying whether the lags of a variable, say a country’s equity returns, are useful when predicting the current outcomes of another variable, say a country’s bond returns. As a result, we have plotted the CCF for each country’s asset-return where we identified that in all our sample countries, lags of one asset show significant ability to predict the outcome of the other asset. However, examining a significant cross-correlation sample does not imply, for example, that the lags of equity returns better explain the current outcome in the bond returns, because we have not accounted for the lags of the bond returns that help to explain the current bond returns. Consequently, the Granger causality test is required to overcome this obstacle.

In the univariate case, we applied the Granger causality test by running two models. The first represents the unrestricted model where one of the asset returns is regressed on its own lag and the lag of the other asset-return while the other model is the asset return is regressed

6_{Perron(1989) argue that in case KPSS test results in rejecting its null hypothesis, it might be misleading to}

infer that the process is not stationary. For example, there might be structural breaks in that data-set that are leading to such inference.

112 Frequency-based Bootstrap Methods for DC Pension Plan Strategy Evaluation

on its lags only. We have observed that in Australia, Ireland, Switzerland, the UK, and the World, both returns of bonds and equities of the aforementioned countries Granger cause one another. Whereas in countries like Denmark, Italy, and South Africa, equity returns Granger causes bond returns. In France, Norway, and Spain bond returns Granger cause equity returns. While the rest of the countries like Belgium, Canada, Germany, Japan, Netherlands, Sweden, and the US neither asset-returns Granger cause one another. Therefore, in the countries where neither asset returns Granger cause the other, accounting for the lags of either asset returns does not help in better explaining the other corresponding current asset-return.

Based on the mentioned analysis, we have tested for ARCH effect on our series accounting for all relevant series that help in predicting it7. For example, testing for ARCH effects for the US equity returns involves the following steps: 1) Fit an ARMA model to this series; 2) extract the residuals of the fitted model; 3) regress their squares on its lags; 4) test the joint significance of the newly formed regression. The result reveals that there exist no ARCH effects in the asset returns for each of the countries Belgium, Canada, Germany, Japan, Netherlands, Sweden, and the US.

Although all the discussed univariate tests reveal important details about individual series, our profound understanding of the properties of the DGP have not reached the optimal point because we have not addressed the joint series distributions and their dynamics. The study of joint series is important, especially when we believe that some of the asset returns in some countries happen to Granger cause one another. Therefore, it is important to treat both country-specific asset returns as being a priori endogenous to avoid some miss-inferences due to the structural cross-correlations between both variables (along with their lags). In the following section, we will continue our analysis in a Multivariate context.